Post on 01-Jun-2020
transcript
Mass in
Kahler Geometry
Claude LeBrunStony Brook University
New Horizons in Twistor TheoryOxford, January 5, 2017
1
Joint work with
2
Joint work with
Hans-Joachim HeinUniversity of Maryland
3
Joint work with
Hans-Joachim HeinFordham University
4
Joint work with
Hans-Joachim HeinFordham University
Comm. Math. Phys. 347 (2016) 621–653.
5
Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn
................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
...............................................................................................................................................................................................
............................................................................................................................................................................................................................................................................
..........................................................................................................................................................................................................................................................................................................................................................................
................................................................................. .....................................................
..................................
..................................
......................................................................................................
...............................................................
.......................................
6
Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn
................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
...............................................................................................................................................................................................
............................................................................................................................................................................................................................................................................
..........................................................................................................................................................................................................................................................................................................................................................................
................................................................................. .....................................................
..................................
..................................
......................................................................................................
...............................................................
.......................................
n ≥ 3
7
Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that
................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
...............................................................................................................................................................................................
............................................................................................................................................................................................................................................................................
..........................................................................................................................................................................................................................................................................................................................................................................
................................................................................. .....................................................
..................................
..................................
......................................................................................................
...............................................................
.......................................
n ≥ 3
8
Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that
................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
...............................................................................................................................................................................................
............................................................................................................................................................................................................................................................................
..........................................................................................................................................................................................................................................................................................................................................................................
................................................................................. .....................................................
..................................
..................................
......................................................................................................
...............................................................
.......................................
n ≥ 3
9
Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that
................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
...............................................................................................................................................................................................
............................................................................................................................................................................................................................................................................
..........................................................................................................................................................................................................................................................................................................................................................................
.......................................
................................................................................. .....................................................
..................................
..................................
......................................................................................................
...............................................................
.......................................
10
Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that
................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
...............................................................................................................................................................................................
.................................................................................................................................................................................................................................................
................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................
.......................................
................................................................................. .....................................................
.....................................................................................................................................................................................
.
..................................
..................................
......................................................................................................
...............................................................
.......................................
..................
.....................
.........................
..............................
.......................................
...........................................................
.............................................................................................................................................................................................................................................................................................................................
............................. ...........
11
Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that
................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
...............................................................................................................................................................................................
.................................................................................................................................................................................................................................................
................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................
.......................................
................................................................................. .....................................................
.....................................................................................................................................................................................
.
..................................
..................................
......................................................................................................
...............................................................
.......................................
..................
.....................
.........................
..............................
.......................................
...........................................................
.............................................................................................................................................................................................................................................................................................................................
............................. ...........
~x
gjk = δjk + O(|x|1−n2−ε)
12
Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that
................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
...............................................................................................................................................................................................
.................................................................................................................................................................................................................................................
................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................
.......................................
................................................................................. .....................................................
.....................................................................................................................................................................................
.
..................................
..................................
......................................................................................................
...............................................................
.......................................
..................
.....................
.........................
..............................
.......................................
...........................................................
.............................................................................................................................................................................................................................................................................................................................
............................. ...........
~x
gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), s ∈ L1
13
Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that
................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
...............................................................................................................................................................................................
.................................................................................................................................................................................................................................................
................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................
.......................................
................................................................................. .....................................................
.....................................................................................................................................................................................
.
..................................
..................................
......................................................................................................
...............................................................
.......................................
..................
.....................
.........................
..............................
.......................................
...........................................................
.............................................................................................................................................................................................................................................................................................................................
............................. ...........
~x
gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), s ∈ L1
14
Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that
................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
...............................................................................................................................................................................................
.................................................................................................................................................................................................................................................
................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................
.......................................
................................................................................. .....................................................
.....................................................................................................................................................................................
.
..................................
..................................
......................................................................................................
...............................................................
.......................................
..................
.....................
.........................
..............................
.......................................
...........................................................
.............................................................................................................................................................................................................................................................................................................................
............................. ...........
~x
gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), scalar curvature ∈ L1
15
Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that
................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
...............................................................................................................................................................................................
.................................................................................................................................................................................................................................................
................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................
.......................................
................................................................................. .....................................................
.....................................................................................................................................................................................
.
..................................
..................................
......................................................................................................
...............................................................
.......................................
..................
.....................
.........................
..............................
.......................................
...........................................................
.............................................................................................................................................................................................................................................................................................................................
............................. ...........
~x
gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), s ∈ L1
16
Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that
................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
...............................................................................................................................................................................................
.................................................................................................................................................................................................................................................
................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................
.......................................
................................................................................. .....................................................
.....................................................................................................................................................................................
.
..................................
..................................
......................................................................................................
...............................................................
.......................................
..................
.....................
.........................
..............................
.......................................
...........................................................
.............................................................................................................................................................................................................................................................................................................................
............................. ...........
~x
gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), s ∈ L1
17
Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
.......
.......
.......
.......
.......
.................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
...............................................................................................................................................................................................
.................................................................................................................................................................................................................................................
................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................................................................
.........................................................................................................................................................
.......................................
.......................................
................................................................................. .....................................................
.....................................................................................................................................................................................
.
..................................
..................................
..................................
..................................
.............................
.............................
..................................
........
..........................
......................................................................................................
...............................................................
.......................................
gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), s ∈ L1
18
Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
.......
.......
.......
.......
.......
.................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
...............................................................................................................................................................................................
.................................................................................................................................................................................................................................................
................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................................................................
.........................................................................................................................................................
.......................................
.......................................
................................................................................. .....................................................
.....................................................................................................................................................................................
.
..................................
..................................
..................................
..................................
.............................
.............................
..................................
........
..........................
......................................................................................................
...............................................................
.......................................
...................................
....................
.....................
.......................
..........................
.............................
.................................
........................................
.................................................
.......................................................................
......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................. ...........
gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), s ∈ L1
19
Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
.......
.......
.......
.......
.......
.................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
...............................................................................................................................................................................................
.................................................................................................................................................................................................................................................
................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................................................................
.........................................................................................................................................................
.......................................
.......................................
................................................................................. .....................................................
.....................................................................................................................................................................................
.
..................................
..................................
..................................
..................................
.............................
.............................
..................................
........
..........................
......................................................................................................
...............................................................
.......................................
..................
.....................
.........................
..............................
.......................................
...........................................................
.............................................................................................................................................................................................................................................................................................................................
............................. ...........
gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), s ∈ L1
20
Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
.......
.......
.......
.......
.......
.................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
...............................................................................................................................................................................................
.................................................................................................................................................................................................................................................
................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................................................................
.........................................................................................................................................................
.......................................
.......................................
................................................................................. .....................................................
.....................................................................................................................................................................................
.
..................................
..................................
..................................
..................................
.............................
.............................
..................................
........
..........................
......................................................................................................
...............................................................
.......................................
...................................
....................
.....................
.......................
..........................
.............................
.................................
........................................
.................................................
.......................................................................
......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................. ...........
gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), s ∈ L1
21
Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
.......
.......
.......
.......
.......
.................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
...............................................................................................................................................................................................
.................................................................................................................................................................................................................................................
................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................................................................
.........................................................................................................................................................
.......................................
.......................................
................................................................................. .....................................................
.....................................................................................................................................................................................
.
..................................
..................................
..................................
..................................
.............................
.............................
..................................
........
..........................
......................................................................................................
...............................................................
.......................................
..................
.....................
.........................
..............................
.......................................
...........................................................
.............................................................................................................................................................................................................................................................................................................................
............................. ...........
gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), s ∈ L1
22
Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each “end” component of is diffeo-morphic to Rn −Dn in such a manner that
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
.......
.......
.......
.......
.......
.................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
...............................................................................................................................................................................................
.................................................................................................................................................................................................................................................
................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................................................................
.........................................................................................................................................................
.......................................
.......................................
................................................................................. .....................................................
.....................................................................................................................................................................................
.
..................................
..................................
..................................
..................................
.............................
.............................
..................................
........
..........................
......................................................................................................
...............................................................
.......................................
..................
.....................
.........................
..............................
.......................................
...........................................................
.............................................................................................................................................................................................................................................................................................................................
............................. ...........
gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), s ∈ L1
23
Definition. Complete, non-compact n-manifold(Mn, g) is asymptotically locally Euclidean (ALE)if ∃ compact set K ⊂ M such that M − K ≈∐i(R
n −Dn)/Γi, where Γi ⊂ O(n), such that
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
...............................................................................................................................................................................................................................................................................................................................
...............................................................................................................................................................................................................................................................................................................................
.........................................................................................................................................................
.........................................................................................................................................................
................................................................................. .....................................................
..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
..................................
..................................
..................................
.............................
.............................
..................................
........
..........................
..................................................
.....................
...............................
24
Definition. Complete, non-compact n-manifold(Mn, g) is asymptotically locally Euclidean (ALE)if ∃ compact set K ⊂ M such that M − K ≈∐i(R
n −Dn)/Γi, where Γi ⊂ O(n), such that
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
...............................................................................................................................................................................................................................................................................................................................
...............................................................................................................................................................................................................................................................................................................................
.........................................................................................................................................................
.........................................................................................................................................................
................................................................................. .....................................................
..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
..................................
..................................
..................................
.............................
.............................
..................................
........
..........................
..................................................
.....................
...............................
25
Definition. Complete, non-compact n-manifold(Mn, g) is asymptotically locally Euclidean (ALE)if ∃ compact set K ⊂ M such that M − K ≈∐i(R
n −Dn)/Γi, where Γi ⊂ O(n), such that
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
...............................................................................................................................................................................................................................................................................................................................
...............................................................................................................................................................................................................................................................................................................................
.........................................................................................................................................................
.........................................................................................................................................................
.......................................
.......................................
................................................................................. .....................................................
..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
..................................
..................................
..................................
.............................
.............................
..................................
........
..........................
..................................................
.....................
...............................
26
Definition. Complete, non-compact n-manifold(Mn, g) is asymptotically locally Euclidean (ALE)if ∃ compact set K ⊂ M such that M − K ≈∐i(Rn −Dn)/Γi, where Γi ⊂ O(n), such that
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
...............................................................................................................................................................................................................................................................................................................................
...............................................................................................................................................................................................................................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.....
.................................................................................................................................................
........................................................
...................................................................................................................................
........................................................
............................
............................
............................
............................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................................................................
.........................................................................................................................................................
.......................................
.......................................
................................................................................. .....................................................
.................................................................
......................................................................................................................
..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
..................................
..................................
..................................
.............................
.............................
..................................
........
..........................
..................................................
.....................
...............................
.............................
................................................
................................................................................................................................................................................................................................... .......
....
27
Definition. Complete, non-compact n-manifold(Mn, g) is asymptotically locally Euclidean (ALE)if ∃ compact set K ⊂ M such that M − K ≈∐i(Rn −Dn)/Γi, where Γi ⊂ O(n), such that
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
...............................................................................................................................................................................................................................................................................................................................
...............................................................................................................................................................................................................................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.....
.................................................................................................................................................
........................................................
...................................................................................................................................
........................................................
............................
............................
............................
............................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................................................................
.........................................................................................................................................................
.......................................
.......................................
................................................................................. .....................................................
.................................................................
......................................................................................................................
..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
..................................
..................................
..................................
.............................
.............................
..................................
........
..........................
..................................................
.....................
...............................
.............................
................................................
................................................................................................................................................................................................................................... .......
....
28
Definition. Complete, non-compact n-manifold(Mn, g) is asymptotically locally Euclidean (ALE)if ∃ compact set K ⊂ M such that M − K ≈∐i(Rn −Dn)/Γi, where Γi ⊂ O(n), such that
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
...............................................................................................................................................................................................................................................................................................................................
...............................................................................................................................................................................................................................................................................................................................
................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................................................................
.........................................................................................................................................................
.......................................
.......................................
................................................................................. .....................................................
.....................................................................................................................................................................................
.
..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
..................................
..................................
..................................
.............................
.............................
..................................
........
..........................
..................................................
.....................
...............................
...................................
....................
.....................
.......................
..........................
.............................
.................................
........................................
.................................................
.......................................................................
......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................. ...........
29
Definition. Complete, non-compact n-manifold(Mn, g) is asymptotically locally Euclidean (ALE)if ∃ compact set K ⊂ M such that M − K ≈∐i(Rn −Dn)/Γi, where Γi ⊂ O(n), such that
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
...............................................................................................................................................................................................................................................................................................................................
...............................................................................................................................................................................................................................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.....
.................................................................................................................................................
........................................................
...................................................................................................................................
........................................................
............................
............................
............................
............................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................................................................
.........................................................................................................................................................
.......................................
.......................................
................................................................................. .....................................................
.................................................................
......................................................................................................................
..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
..................................
..................................
..................................
.............................
.............................
..................................
........
..........................
..................................................
.....................
...............................
.............................
................................................
................................................................................................................................................................................................................................... .......
....
30
Definition. Complete, non-compact n-manifold(Mn, g) is asymptotically locally Euclidean (ALE)if ∃ compact set K ⊂ M such that M − K ≈∐i(Rn −Dn)/Γi, where Γi ⊂ O(n), such that
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
...............................................................................................................................................................................................................................................................................................................................
...............................................................................................................................................................................................................................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.....
.................................................................................................................................................
........................................................
...................................................................................................................................
........................................................
............................
............................
............................
............................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................................................................
.........................................................................................................................................................
.......................................
.......................................
................................................................................. .....................................................
.................................................................
......................................................................................................................
..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
..................................
..................................
..................................
.............................
.............................
..................................
........
..........................
..................................................
.....................
...............................
.............................
................................................
................................................................................................................................................................................................................................... .......
....
gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), s ∈ L1
31
Why consider ALE spaces?
32
Key examples:
33
Key examples:
Term ALE coined by Gibbons & Hawking, 1979.
34
Key examples:
Term ALE coined by Gibbons & Hawking, 1979.
They wrote down various explicit Ricci-flat ALE4-manifolds they called gravitational instantons.
35
Key examples:
Term ALE coined by Gibbons & Hawking, 1979.
They wrote down various explicit Ricci-flat ALE4-manifolds they called gravitational instantons.
By contrast, any Ricci-flat AE manifold must beflat, by the Bishop-Gromov inequality. . .
36
Key examples:
Term ALE coined by Gibbons & Hawking, 1979.
They wrote down various explicit Ricci-flat ALE4-manifolds they called gravitational instantons.
Their examples have just one end, with
37
Key examples:
Term ALE coined by Gibbons & Hawking, 1979.
They wrote down various explicit Ricci-flat ALE4-manifolds they called gravitational instantons.
Their examples have just one end, with
Γ ∼= Z` ⊂ SU(2) ⊂ O(4).
38
vv
vvv
Data: ` points in R3. =⇒ V with ∆V = 0
39
vv
vvv
Data: ` points in R3. =⇒ V with ∆V = 0
V =∑j=1
1
2%j
40
vv
vvv
.......
.......
........................................................
..............................
.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................
.......
.......
........................................................
..............................
.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................
.......
.......
........................................................
..............................
.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................
.......
.......
........................................................
..............................
.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................
.......
.......
........................................................
..............................
.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................
Data: ` points in R3. =⇒ V with ∆V = 0
V =∑j=1
1
2%j
F = ?dV curvature θ on P → R3 − pts.
41
vv
vvv
.......
.......
........................................................
..............................
.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................
.......
.......
........................................................
..............................
.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................
.......
.......
........................................................
..............................
.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................
.......
.......
........................................................
..............................
.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................
.......
.......
........................................................
..............................
.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................
Data: ` points in R3. =⇒ V with ∆V = 0
g = V h + V −1θ2
F = ?dV curvature θ on P → R3 − pts.
42
vv
vvv
.......
.......
........................................................
..............................
.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................
.......
.......
........................................................
..............................
.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................
.......
.......
........................................................
..............................
.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................
.......
.......
........................................................
..............................
.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................
.......
.......
........................................................
..............................
.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................
Data: ` points in R3. =⇒ V with ∆V = 0
g = V h + V −1θ2
on P . Then take M4 = Riemannian completion.
43
Key examples:
Term ALE coined by Gibbons & Hawking, 1979.
They wrote down various explicit Ricci-flat ALE4-manifolds they called gravitational instantons.
Their examples have just one end, with
Γ ∼= Z` ⊂ SU(2) ⊂ O(4).
44
Key examples:
Term ALE coined by Gibbons & Hawking, 1979.
They wrote down various explicit Ricci-flat ALE4-manifolds they called gravitational instantons.
Their examples have just one end, with
Γ ∼= Z` ⊂ SU(2) ⊂ O(4).
The G-H metrics are hyper-Kahler, and were soonrediscovered independently by Hitchin.
45
Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
46
Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
s
...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
...................................................
....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................
...........................................................
...................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................
47
Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
s
...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
...................................................
....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................
...........................................................
...................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................
..........................................
..................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................
48
Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
s
...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
...................................................
....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................
...........................................................
...................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................
..........................................
..................
............................................
.........................................
..............
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................
49
Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
s
...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
...................................................
....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................
...........................................................
...................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................
..........................................
..................
............................................
.........................................
..............
................................................
........................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................
50
Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
s
...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
...................................................
....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................
...........................................................
...................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................
..........................................
..................
............................................
.........................................
..............
................................................
........................................................
............................................
.........................................
..............
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................
51
Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
s
...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
...................................................
....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................
...........................................................
...................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................
..........................................
..................
............................................
.........................................
..............
................................................
........................................................
............................................
.........................................
..............
........................................
..........................................
..................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................
52
Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
s
...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
...................................................
....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................
...........................................................
...................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................
..........................................
..................
............................................
.........................................
..............
................................................
........................................................
............................................
.........................................
..............
........................................
..........................................
..................
..............................................................................
...................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................
53
Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
s
...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
...................................................
....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................
...........................................................
...................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................
..........................................
..................
............................................
.........................................
..............
................................................
........................................................
............................................
.........................................
..............
........................................
..........................................
..................
..............................................................................
...................
.........................................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................
54
Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
s
...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
...................................................
....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................
...........................................................
...................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................
..........................................
..................
............................................
.........................................
..............
................................................
........................................................
............................................
.........................................
..............
........................................
..........................................
..................
..............................................................................
...................
.........................................................................................................................
................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................
55
Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
s
...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
...................................................
....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................
...........................................................
...................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................
..........................................
..................
............................................
.........................................
..............
................................................
........................................................
............................................
.........................................
..............
........................................
..........................................
..................
..............................................................................
...................
.........................................................................................................................
................................................................................................
....................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................
56
Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
s
...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
...................................................
....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................
...........................................................
...................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................
..........................................
..................
............................................
.........................................
..............
................................................
........................................................
............................................
.........................................
..............
........................................
..........................................
..................
..............................................................................
...................
.........................................................................................................................
................................................................................................
....................................................................................................
...................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................
57
Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
s
...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
...................................................
....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................
...........................................................
...................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................
..........................................
..................
............................................
.........................................
..............
................................................
........................................................
............................................
.........................................
..............
........................................
..........................................
..................
..............................................................................
...................
.........................................................................................................................
................................................................................................
....................................................................................................
...................................................................................................
..................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................
58
Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
s
...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
...................................................
....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................
...........................................................
...................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................
..........................................
..................
............................................
.........................................
..............
................................................
........................................................
............................................
.........................................
..............
........................................
..........................................
..................
..............................................................................
...................
.........................................................................................................................
................................................................................................
....................................................................................................
...................................................................................................
..................................................................................................
......................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................
59
Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
s
...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
...................................................
....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................
...........................................................
...................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................
..........................................
..................
............................................
.........................................
..............
................................................
........................................................
............................................
.........................................
..............
........................................
..........................................
..................
..............................................................................
...................
.........................................................................................................................
................................................................................................
....................................................................................................
...................................................................................................
..................................................................................................
......................................................................................................
.....................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................
60
Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
s
...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
...................................................
....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................
...........................................................
...................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................................
........................................
........................................
.....................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................
61
Hyper-Kahler metrics:
(M4, g) hyper-Kahler ⇐⇒ holonomy ⊂ Sp(1)
s
...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
...................................................
....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................
...........................................................
...................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................................
........................................
........................................
.....................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................
62
Hyper-Kahler metrics:
(M4, g) hyper-Kahler ⇐⇒ holonomy ⊂ Sp(1)
s
...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
...................................................
....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................
...........................................................
...................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................................
........................................
........................................
.....................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................
Sp(1) = SU(2)
63
Hyper-Kahler metrics:
(M4, g) hyper-Kahler ⇐⇒ holonomy ⊂ Sp(1)
s
...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
...................................................
....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................
...........................................................
...................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................................
........................................
........................................
.....................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................
Sp(1) = SU(2)
⇐⇒ Λ+ flat and trivial.
64
Hyper-Kahler metrics:
(M4, g) hyper-Kahler ⇐⇒ holonomy ⊂ Sp(1)
s
...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
...................................................
....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................
...........................................................
...................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................................
........................................
........................................
.....................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................
Sp(1) = SU(2)
⇐⇒ Λ+ flat and trivial.
Locally, ⇐⇒ s = 0, r = 0, W+ = 0.
65
Hyper-Kahler metrics:
(M4, g) hyper-Kahler ⇐⇒ holonomy ⊂ Sp(1)
s
...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
...................................................
....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................
...........................................................
...................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................................
........................................
........................................
.....................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................
Sp(1) = SU(2)
Ricci-flat and Kahler,
for many different complex structures!
66
Hyper-Kahler metrics:
(M4, g) hyper-Kahler ⇐⇒ holonomy ⊂ U(2)
s
...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
...................................................
....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................
...........................................................
...................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................................
........................................
........................................
.....................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................
Sp(1) ⊂ U(2)
Ricci-flat and Kahler,
for many different complex structures!
67
Hyper-Kahler metrics:
(M4, g) hyper-Kahler ⇐⇒ holonomy ⊂ Sp(1)
s
...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
...................................................
....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................
...........................................................
...................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................................
........................................
........................................
.....................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................
Sp(1) = SU(2)
Ricci-flat and Kahler,
for many different complex structures!
68
All these complex structures can be repackaged as
Penrose Twistor Space (Z6, J),
which is a complex 3-manifold
.....................................................................................................................................................................................................................................................................................................................................................................................................................................
............................
.......................
.......................................................................................................................................................
................. .................................................................................................
...........................
.........................
...........................................
....................................r
.............
..........
......................................
...................................
........................
......................
r
r Z
M 4
↓
.......................................................................
..................
..................
..................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
......................................................................................
.......................................................................
..................
..................
..................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.......................................................................
..................
..................
..................
.................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.......................................................................
..................
..................
..................
.................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.......................................................................
..................
..................
..................
.................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.......................................................................
..................
..................
..................
.................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................................................................ .................
......................................................................................
.......................................................................
............................... ...............................
..................................................................................................................................................
......................................................................................................................
.....................................................
......................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................
.............................................................................
...............................................................................................................................................................................................................................................................................................................................................................................................................
69
All these complex structures can be repackaged as
Penrose Twistor Space (Z, J),
which is a complex 3-manifold
.....................................................................................................................................................................................................................................................................................................................................................................................................................................
............................
.......................
.......................................................................................................................................................
................. .................................................................................................
...........................
.........................
...........................................
....................................r
.............
..........
......................................
...................................
........................
......................
r
r Z
M 4
↓
.......................................................................
..................
..................
..................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
......................................................................................
.......................................................................
..................
..................
..................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.......................................................................
..................
..................
..................
.................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.......................................................................
..................
..................
..................
.................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.......................................................................
..................
..................
..................
.................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.......................................................................
..................
..................
..................
.................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................................................................ .................
......................................................................................
.......................................................................
............................... ...............................
..................................................................................................................................................
......................................................................................................................
.....................................................
......................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................
.............................................................................
...............................................................................................................................................................................................................................................................................................................................................................................................................
70
All these complex structures can be repackaged as
Penrose Twistor Space (Z, J),
which is a complex 3-manifold.
.....................................................................................................................................................................................................................................................................................................................................................................................................................................
............................
.......................
.......................................................................................................................................................
................. .................................................................................................
...........................
.........................
...........................................
....................................r
.............
..........
......................................
...................................
........................
......................
r
r Z
M 4
↓
.......................................................................
..................
..................
..................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
......................................................................................
.......................................................................
..................
..................
..................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.......................................................................
..................
..................
..................
.................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.......................................................................
..................
..................
..................
.................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.......................................................................
..................
..................
..................
.................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.......................................................................
..................
..................
..................
.................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................................................................ .................
......................................................................................
.......................................................................
............................... ...............................
..................................................................................................................................................
......................................................................................................................
.....................................................
......................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................
.............................................................................
...............................................................................................................................................................................................................................................................................................................................................................................................................
71
All these complex structures can be repackaged as
Penrose Twistor Space (Z, J),
which is a complex 3-manifold.
.....................................................................................................................................................................................................................................................................................................................................................................................................................................
............................
.......................
.......................................................................................................................................................
................. .................................................................................................
...........................
.........................
...........................................
....................................r
.............
..........
......................................
...................................
........................
......................
r
r Z
M 4
↓
.......................................................................
..................
..................
..................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
......................................................................................
.......................................................................
..................
..................
..................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.......................................................................
..................
..................
..................
.................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.......................................................................
..................
..................
..................
.................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.......................................................................
..................
..................
..................
.................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.......................................................................
..................
..................
..................
.................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................................................................ .................
......................................................................................
.......................................................................
............................... ...............................
..................................................................................................................................................
......................................................................................................................
.....................................................
......................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................
.............................................................................
...............................................................................................................................................................................................................................................................................................................................................................................................................
Riemannian non-linear graviton construction.
72
Hitchin’s Twistor Spaces:
73
Hitchin’s Twistor Spaces:
H0(CP1,O(2)) = C3⊃ R3.
74
Hitchin’s Twistor Spaces:
H0(CP1,O(2)) = C3 ⊃ R3.
75
Hitchin’s Twistor Spaces:
H0(CP1,O(2)) = C3 ⊃ R3.
vv
vvv
So ` points determine P1, . . . , P` ∈ H0(CP1,O(2)).
76
Hitchin’s Twistor Spaces:
H0(CP1,O(2)) = C3 ⊃ R3.
vv
vvv
So ` points determine P1, . . . , P` ∈ H0(CP1,O(2)).
Small resolution Z of Z ⊂ O(`)⊕O(`)⊕O(2)
77
Hitchin’s Twistor Spaces:
H0(CP1,O(2)) = C3 ⊃ R3.
vv
vvv
So ` points determine P1, . . . , P` ∈ H0(CP1,O(2)).
Small resolution Z of Z ⊂ O(`)⊕O(`)⊕O(2)
xy = (z − P1) · · · (z − P`)
78
Hitchin’s Twistor Spaces:
H0(CP1,O(2)) = C3 ⊃ R3.
vv
vvv
So ` points determine P1, . . . , P` ∈ H0(CP1,O(2)).
Small resolution Z of Z ⊂ O(`)⊕O(`)⊕O(2)
xy = (z − P1) · · · (z − P`)
79
Hitchin’s Twistor Spaces:
H0(CP1,O(2)) = C3 ⊃ R3.
vv
vvv
So ` points determine P1, . . . , P` ∈ H0(CP1,O(2)).
Small resolution Z of Z ⊂ O(`)⊕O(`)⊕O(2)
xy = (z − P1) · · · (z − P`)is the twistor space of a Gibbons-Hawking metric.
80
Key examples:
Term ALE coined by Gibbons & Hawking, 1979.
They wrote down various explicit Ricci-flat ALE4-manifolds they called gravitational instantons.
Their examples have just one end, with
Γ ∼= Z` ⊂ SU(2) ⊂ O(4).
The G-H metrics are hyper-Kahler, and were soonrediscovered independently by Hitchin.
81
Key examples:
Term ALE coined by Gibbons & Hawking, 1979.
They wrote down various explicit Ricci-flat ALE4-manifolds they called gravitational instantons.
Their examples have just one end, with
Γ ∼= Z` ⊂ SU(2) ⊂ O(4).
The G-H metrics are hyper-Kahler, and were soonrediscovered independently by Hitchin.
Hitchin conjectured that similar metrics would existfor each finite Γ ⊂ SU(2).
82
Key examples:
Term ALE coined by Gibbons & Hawking, 1979.
They wrote down various explicit Ricci-flat ALE4-manifolds they called gravitational instantons.
Their examples have just one end, with
Γ ∼= Z` ⊂ SU(2) ⊂ O(4).
The G-H metrics are hyper-Kahler, and were soonrediscovered independently by Hitchin.
Hitchin conjectured that similar metrics would existfor each finite Γ ⊂ SU(2).
This conjecture was proved by Kronheimer, 1986.
83
Felix Klein, 1884: C2/Γ → C3
Zk+1 ←→ xy + zk+1 = 0
Dih∗k−2 ←→ x2 + z(y2 + zk−2) = 0
T ∗ ←→ x2 + y3 + z4 = 0
O∗ ←→ x2 + y3 + yz3 = 0
I∗ ←→ x2 + y3 + z5 = 0
84
Zk+1←→Ak • • • •.......................................................................................................................................................
Dih∗k−2←→Dk • • • ••.................................................................................................................
................................................
...........................................................
T ∗←→E6 • • • •••................................................................................................................................................................................................................
.......
.......
.......
.......
.......
....
O∗←→E7 • • • ••• •..................................................................................................................................................................................................................................................................
.......
.......
.......
.......
.......
....
I∗←→E8 • • • ••• • •....................................................................................................................................................................................................................................................................................................................
.......
.......
.......
.......
.......
....
85
Hyper-Kahler metrics:
(M4, g) hyper-Kahler ⇐⇒ holonomy ⊂ U(2)
s
...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
...................................................
....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................
...........................................................
...................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................................
........................................
........................................
.....................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................
86
Hyper-Kahler metrics:
(M4, g) hyper-Kahler ⇐⇒ holonomy ⊂ U(2)
s
...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
...................................................
....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................
...........................................................
...................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................................
........................................
........................................
.....................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................
In real dimension 4, any Kahler manifold satisfies
|W+|2 =s2
24
87
Hyper-Kahler metrics:
(M4, g) hyper-Kahler ⇐⇒ holonomy ⊂ U(2)
s
...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
...................................................
....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................
...........................................................
...................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................................
........................................
........................................
.....................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................
In real dimension 4, any Kahler manifold satisfies
|W+|2 =s2
24
so that W+ = 0 ⇐⇒ s = 0.
88
Any scalar-flat Kahler surface (M4, g, J) has a
Penrose Twistor Space (Z, J),
which is once again a complex 3-manifold.
.....................................................................................................................................................................................................................................................................................................................................................................................................................................
............................
.......................
.........................
.......................................................................................................................................................
.................
.............
..........
......................................
...................................
........................
.................................................................
....................................
r
rr Z
M 4
↓
.......................................................................
..................
..................
..................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.............................................................................................................................................................
..................
..................
..................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................................................................ .................
......................................................................................
.......................................................................
............................... ...............................
..................................................................................................................................................
......................................................................................................................
.....................................................
......................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................
.............................................................................
...............................................................................................................................................................................................................................................................................................................................................................................................................
89
Any scalar-flat Kahler surface (M4, g, J) has a
Penrose Twistor Space (Z, J),
which is once again a complex 3-manifold.
.....................................................................................................................................................................................................................................................................................................................................................................................................................................
............................
.......................
.........................
.......................................................................................................................................................
.................
.............
..........
......................................
...................................
........................
.................................................................
....................................
r
rr Z
M 4
↓
.......................................................................
..................
..................
..................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.............................................................................................................................................................
..................
..................
..................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................................................................ .................
......................................................................................
.......................................................................
............................... ...............................
..................................................................................................................................................
......................................................................................................................
.....................................................
......................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................
.............................................................................
...............................................................................................................................................................................................................................................................................................................................................................................................................
Kahler case: holomorphic section of KZ−1/2
vanishing at D := (M,J) ∩ (M,−J).
90
Any scalar-flat Kahler surface (M4, g, J) has a
Penrose Twistor Space (Z, J),
which is once again a complex 3-manifold.
.....................................................................................................................................................................................................................................................................................................................................................................................................................................
............................
.......................
.........................
.......................................................................................................................................................
.................
.............
..........
......................................
...................................
........................
.................................................................
....................................
r
rr Z
M 4
↓
.......................................................................
..................
..................
..................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.............................................................................................................................................................
..................
..................
..................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................................................................ .................
......................................................................................
.......................................................................
............................... ...............................
..................................................................................................................................................
......................................................................................................................
.....................................................
......................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................
.............................................................................
...............................................................................................................................................................................................................................................................................................................................................................................................................
Kahler case: holomorphic section of KZ−1/2
vanishing at D := (M,J) ∩ (M,−J).
=⇒ Normal bundle of (M,J) in Z is KM−1.
91
Any scalar-flat Kahler surface (M4, g, J) has a
Penrose Twistor Space (Z, J),
which is once again a complex 3-manifold.
.....................................................................................................................................................................................................................................................................................................................................................................................................................................
............................
.......................
.........................
.......................................................................................................................................................
.................
.............
..........
......................................
...................................
........................
.................................................................
....................................
r
rr Z
M 4
↓
.......................................................................
..................
..................
..................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.............................................................................................................................................................
..................
..................
..................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................................................................ .................
......................................................................................
.......................................................................
............................... ...............................
..................................................................................................................................................
......................................................................................................................
.....................................................
......................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................
.............................................................................
...............................................................................................................................................................................................................................................................................................................................................................................................................
0→ T 1,0M → T 1,0Z|M → KM−1→ 0
=⇒ Normal bundle of (M,J) in Z is KM−1.
92
Any scalar-flat Kahler surface (M4, g, J) has a
Penrose Twistor Space (Z, J),
which is once again a complex 3-manifold.
.....................................................................................................................................................................................................................................................................................................................................................................................................................................
............................
.......................
.........................
.......................................................................................................................................................
.................
.............
..........
......................................
...................................
........................
.................................................................
....................................
r
rr Z
M 4
↓
.......................................................................
..................
..................
..................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.............................................................................................................................................................
..................
..................
..................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................................................................ .................
......................................................................................
.......................................................................
............................... ...............................
..................................................................................................................................................
......................................................................................................................
.....................................................
......................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................
.............................................................................
...............................................................................................................................................................................................................................................................................................................................................................................................................
0→ T 1,0M → T 1,0Z|M → KM−1→ 0
Extension class: [ω] ∈ H1(M,O(KM⊗T 1,0M)).
93
Any scalar-flat Kahler surface (M4, g, J) has a
Penrose Twistor Space (Z, J),
which is once again a complex 3-manifold.
.....................................................................................................................................................................................................................................................................................................................................................................................................................................
............................
.......................
.........................
.......................................................................................................................................................
.................
.............
..........
......................................
...................................
........................
.................................................................
....................................
r
rr Z
M 4
↓
.......................................................................
..................
..................
..................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.............................................................................................................................................................
..................
..................
..................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................................................................ .................
......................................................................................
.......................................................................
............................... ...............................
..................................................................................................................................................
......................................................................................................................
.....................................................
......................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................
.............................................................................
...............................................................................................................................................................................................................................................................................................................................................................................................................
0→ T 1,0M → T 1,0Z|M → KM−1→ 0
Extension class: [ω] ∈ H1(M,Ω1).
94
Any scalar-flat Kahler surface (M4, g, J) has a
Penrose Twistor Space (Z, J),
which is once again a complex 3-manifold.
.....................................................................................................................................................................................................................................................................................................................................................................................................................................
............................
.......................
.........................
.......................................................................................................................................................
.................
.............
..........
......................................
...................................
........................
.................................................................
....................................
r
rr Z
M 4
↓
.......................................................................
..................
..................
..................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.............................................................................................................................................................
..................
..................
..................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................................................................ .................
......................................................................................
.......................................................................
............................... ...............................
..................................................................................................................................................
......................................................................................................................
.....................................................
......................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................
.............................................................................
...............................................................................................................................................................................................................................................................................................................................................................................................................
0→ T 1,0M → T 1,0Z|M → KM−1→ 0
Extension class: [ω] ∈ H1(M,Ω1).
95
Any scalar-flat Kahler surface (M4, g, J) has a
Penrose Twistor Space (Z, J),
which is once again a complex 3-manifold.
.....................................................................................................................................................................................................................................................................................................................................................................................................................................
............................
.......................
.........................
.......................................................................................................................................................
.................
.............
..........
......................................
...................................
........................
.................................................................
....................................
r
rr Z
M 4
↓
.......................................................................
..................
..................
..................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.............................................................................................................................................................
..................
..................
..................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................................................................ .................
......................................................................................
.......................................................................
............................... ...............................
..................................................................................................................................................
......................................................................................................................
.....................................................
......................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................
.............................................................................
...............................................................................................................................................................................................................................................................................................................................................................................................................
0→ T 1,0M → T 1,0Z|M → KM−1→ 0
Kahler class: [ω] ∈ H1(M,Ω1).
96
Any scalar-flat Kahler surface (M4, g, J) has a
Penrose Twistor Space (Z, J),
which is once again a complex 3-manifold.
.....................................................................................................................................................................................................................................................................................................................................................................................................................................
............................
.......................
.........................
.......................................................................................................................................................
.................
.............
..........
......................................
...................................
........................
.................................................................
....................................
r
rr Z
M 4
↓
.......................................................................
..................
..................
..................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.............................................................................................................................................................
..................
..................
..................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................................................................ .................
......................................................................................
.......................................................................
............................... ...............................
..................................................................................................................................................
......................................................................................................................
.....................................................
......................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................
.............................................................................
...............................................................................................................................................................................................................................................................................................................................................................................................................
0→ T 1,0M → T 1,0Z|M → KM−1→ 0
Kahler form: ω = g(J ·, ·).
97
Some AE/ALE Scalar-Flat Kahler Surfaces:
98
Some AE/ALE Scalar-Flat Kahler Surfaces:
(L ’91)
99
Some AE/ALE Scalar-Flat Kahler Surfaces:
100
Some AE/ALE Scalar-Flat Kahler Surfaces:
f
vv
vvv
.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
............................................
...................................
..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................
....................
......................
.......................
.........................
...........................
...............................
......................................
....................................................
..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.....................................................................................................
...............................................................
..............................................
....................................
....................................................................................
...................
............. .............
..........................
............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ..........................
.............
Data: k + 1 points in H3. =⇒ V with ∆V = 0
101
Some AE/ALE Scalar-Flat Kahler Surfaces:
f
vv
vvv
.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
............................................
...................................
..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................
....................
......................
.......................
.........................
...........................
...............................
......................................
....................................................
..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.....................................................................................................
...............................................................
..............................................
....................................
....................................................................................
...................
............. .............
..........................
............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ..........................
.............
Data: k + 1 points in H3. =⇒ V with ∆V = 0
V = 1 +`
e2%0 − 1+
k∑j=1
1
e2%j − 1
102
Some AE/ALE Scalar-Flat Kahler Surfaces:
f
vv
vvv
.......
........................................
..............................................................................................................................................................................................................................................................................................................................................................................
..........
.......
........................................
..............................................................................................................................................................................................................................................................................................................................................................................
..........
.......
........................................
..............................................................................................................................................................................................................................................................................................................................................................................
..........
.......
........................................
..............................................................................................................................................................................................................................................................................................................................................................................
..........
.......
....................................
..............................................................................................................................................................................................................................................................................
....
.......
........................................
..............................................................................................................................................................................................................................................................................................................................................................................
..........
.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
............................................
...................................
..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................
....................
......................
.......................
.........................
...........................
...............................
......................................
....................................................
..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.....................................................................................................
...............................................................
..............................................
....................................
....................................................................................
...................
............. .............
..........................
............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ..........................
.............
Data: k + 1 points in H3. =⇒ V with ∆V = 0
F = ?dV curvature θ on P → H3 − pts.
103
Some AE/ALE Scalar-Flat Kahler Surfaces:
f
vv
vvv
.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
............................................
...................................
..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................
....................
......................
.......................
.........................
...........................
...............................
......................................
....................................................
..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.....................................................................................................
...............................................................
..............................................
....................................
....................................................................................
...................
............. .............
..........................
............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ..........................
.............
Data: k + 1 points in H3. =⇒ V with ∆V = 0
g =1
4 sinh2 %0
(V h + V −1θ2
)
104
Some AE/ALE Scalar-Flat Kahler Surfaces:
f
vv
vvv
.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
............................................
...................................
..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................
....................
......................
.......................
.........................
...........................
...............................
......................................
....................................................
..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.....................................................................................................
...............................................................
..............................................
....................................
....................................................................................
...................
............. .............
..........................
............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ..........................
.............
Riemannian completion is ALE scalar-flat Kahler.
g =1
4 sinh2 %0
(V h + V −1θ2
)
105
Some AE/ALE Scalar-Flat Kahler Surfaces:
f
vv
vvv
.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
............................................
...................................
..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................
....................
......................
.......................
.........................
...........................
...............................
......................................
....................................................
..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.....................................................................................................
...............................................................
..............................................
....................................
....................................................................................
...................
............. .............
..........................
............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ..........................
.............
Riemannian completion is AE ⇐⇒ ` = 1:
V = 1 +`
e2%0 − 1+
k∑j=1
1
e2%j − 1
106
Twistor Spaces for These Metrics:
H0(CP1 × CP1,O(1, 1)) = C4⊃ R1,3 ⊃ H3
107
Twistor Spaces for These Metrics:
H0(CP1 × CP1,O(1, 1)) = C4 ⊃ R1,3⊃ H3
108
Twistor Spaces for These Metrics:
H0(CP1 × CP1,O(1, 1)) = C4 ⊃ R1,3 ⊃ H3
109
Twistor Spaces for These Metrics:
H0(CP1 × CP1,O(1, 1)) = C4 ⊃ R1,3 ⊃ H3
f
vv
vvv
.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
............................................
...................................
..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................
....................
......................
.......................
.........................
...........................
...............................
......................................
....................................................
..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.....................................................................................................
...............................................................
..............................................
....................................
....................................................................................
...................
............. .............
..........................
............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ..........................
.............
So k + 1 points in H3 give rise to
110
Twistor Spaces for These Metrics:
H0(CP1 × CP1,O(1, 1)) = C4 ⊃ R1,3 ⊃ H3
f
vv
vvv
.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
............................................
...................................
..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................
....................
......................
.......................
.........................
...........................
...............................
......................................
....................................................
..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.....................................................................................................
...............................................................
..............................................
....................................
....................................................................................
...................
............. .............
..........................
............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ..........................
.............
So k + 1 points in H3 give rise to
P0, P1, . . . , Pk ∈ H0(CP1 × CP1,O(1, 1)).
111
Twistor Spaces for These Metrics:
H0(CP1 × CP1,O(1, 1)) = C4 ⊃ R1,3 ⊃ H3
112
Twistor Spaces for These Metrics:
H0(CP1 × CP1,O(1, 1)) = C4 ⊃ R1,3 ⊃ H3
InO(k + `− 1, 1)⊕O(1, k + `− 1)→ CP1×CP1,
113
Twistor Spaces for These Metrics:
H0(CP1 × CP1,O(1, 1)) = C4 ⊃ R1,3 ⊃ H3
InO(k + `− 1, 1)⊕O(1, k + `− 1)→ CP1×CP1,
let Z be the hypersurface
xy = P `0 P1 · · · Pk.
114
Twistor Spaces for These Metrics:
H0(CP1 × CP1,O(1, 1)) = C4 ⊃ R1,3 ⊃ H3
InO(k + `− 1, 1)⊕O(1, k + `− 1)→ CP1×CP1,
let Z be the hypersurface
xy = P `0 P1 · · · Pk.
Then twistor space Z obtained from Z by
115
Twistor Spaces for These Metrics:
H0(CP1 × CP1,O(1, 1)) = C4 ⊃ R1,3 ⊃ H3
InO(k + `− 1, 1)⊕O(1, k + `− 1)→ CP1×CP1,
let Z be the hypersurface
xy = P `0 P1 · · · Pk.
Then twistor space Z obtained from Z by
• removing curve in zero section cut out by P0,
116
Twistor Spaces for These Metrics:
H0(CP1 × CP1,O(1, 1)) = C4 ⊃ R1,3 ⊃ H3
InO(k + `− 1, 1)⊕O(1, k + `− 1)→ CP1×CP1,
let Z be the hypersurface
xy = P `0 P1 · · · Pk.
Then twistor space Z obtained from Z by
• removing curve in zero section cut out by P0,
• adding two rational curves at infinity, and
117
Twistor Spaces for These Metrics:
H0(CP1 × CP1,O(1, 1)) = C4 ⊃ R1,3 ⊃ H3
InO(k + `− 1, 1)⊕O(1, k + `− 1)→ CP1×CP1,
let Z be the hypersurface
xy = P `0 P1 · · · Pk.
Then twistor space Z obtained from Z by
• removing curve in zero section cut out by P0,
• adding two rational curves at infinity, and
•making small resolutions of isolated singularities.
118
Any scalar-flat Kahler surface (M4, g, J) has a
Penrose Twistor Space (Z, J),
which is once again a complex 3-manifold.
.....................................................................................................................................................................................................................................................................................................................................................................................................................................
............................
.......................
.........................
.......................................................................................................................................................
.................
.............
..........
......................................
...................................
........................
.................................................................
....................................
r
rr Z
M 4
↓
.......................................................................
..................
..................
..................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.............................................................................................................................................................
..................
..................
..................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................................................................ .................
......................................................................................
.......................................................................
............................... ...............................
..................................................................................................................................................
......................................................................................................................
.....................................................
......................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................
.............................................................................
...............................................................................................................................................................................................................................................................................................................................................................................................................
119
Any scalar-flat Kahler surface (M4, g, J) has a
Penrose Twistor Space (Z, J),
which is once again a complex 3-manifold.
.....................................................................................................................................................................................................................................................................................................................................................................................................................................
............................
.......................
.........................
.......................................................................................................................................................
.................
.............
..........
......................................
...................................
........................
.................................................................
....................................
r
rr Z
M 4
↓
.......................................................................
..................
..................
..................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.............................................................................................................................................................
..................
..................
..................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................................................................ .................
......................................................................................
.......................................................................
............................... ...............................
..................................................................................................................................................
......................................................................................................................
.....................................................
......................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................
.............................................................................
...............................................................................................................................................................................................................................................................................................................................................................................................................
Lots more ALE scalar-flat Kahler surfaces now known:
120
Any scalar-flat Kahler surface (M4, g, J) has a
Penrose Twistor Space (Z, J),
which is once again a complex 3-manifold.
.....................................................................................................................................................................................................................................................................................................................................................................................................................................
............................
.......................
.........................
.......................................................................................................................................................
.................
.............
..........
......................................
...................................
........................
.................................................................
....................................
r
rr Z
M 4
↓
.......................................................................
..................
..................
..................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.............................................................................................................................................................
..................
..................
..................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................................................................ .................
......................................................................................
.......................................................................
............................... ...............................
..................................................................................................................................................
......................................................................................................................
.....................................................
......................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................
.............................................................................
...............................................................................................................................................................................................................................................................................................................................................................................................................
Lots more ALE scalar-flat Kahler surfaces now known:
Joyce, Calderbank-Singer, Lock-Viaclovsky. . .
121
Any scalar-flat Kahler surface (M4, g, J) has a
Penrose Twistor Space (Z, J),
which is once again a complex 3-manifold.
.....................................................................................................................................................................................................................................................................................................................................................................................................................................
............................
.......................
.........................
.......................................................................................................................................................
.................
.............
..........
......................................
...................................
........................
.................................................................
....................................
r
rr Z
M 4
↓
.......................................................................
..................
..................
..................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.............................................................................................................................................................
..................
..................
..................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................................................................ .................
......................................................................................
.......................................................................
............................... ...............................
..................................................................................................................................................
......................................................................................................................
.....................................................
......................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................
.............................................................................
...............................................................................................................................................................................................................................................................................................................................................................................................................
Lots more ALE scalar-flat Kahler surfaces now known:
Joyce, Calderbank-Singer, Lock-Viaclovsky. . .
But full classification remains an open problem.
122
Definition. Complete, non-compact n-manifold(Mn, g) is asymptotically locally Euclidean (ALE)if ∃ compact set K ⊂ M such that M − K ≈∐i(Rn −Dn)/Γi, where Γi ⊂ O(n), such that
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
............................
............................
............................
............................
...............................................................................................................................................................................................................................................................................................................................
...............................................................................................................................................................................................................................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.....
.................................................................................................................................................
........................................................
...................................................................................................................................
........................................................
............................
............................
............................
............................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................................................................
.........................................................................................................................................................
.......................................
.......................................
................................................................................. .....................................................
.................................................................
......................................................................................................................
..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
..................................
..................................
..................................
.............................
.............................
..................................
........
..........................
..................................................
.....................
...............................
.............................
................................................
................................................................................................................................................................................................................................... .......
....
gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), s ∈ L1
123
Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
124
Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
125
Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
126
Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
127
Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
128
Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
where
129
Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
...............................................................................................................................................................................................................................................................................................................................
...............................................................................................................................................................................................................................................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................. .....................................................
..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
..................................
..................................................
.....................
...............................
130
Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
...............................................................................................................................................................................................................................................................................................................................
...............................................................................................................................................................................................................................................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................. .....................................................
..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
..................................
..................................................
.....................
...............................
131
Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
• ν is the outpointing Euclidean unit normal;and
132
Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
• ν is the outpointing Euclidean unit normal;and
• αE is the volume (n− 1)-form induced by theEuclidean metric.
133
Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
• ν is the outpointing Euclidean unit normal;and
• αE is the volume (n− 1)-form induced by theEuclidean metric.
Seems to depend on choice of coordinates!
134
Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
• ν is the outpointing Euclidean unit normal;and
• αE is the volume (n− 1)-form induced by theEuclidean metric.
135
Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
• ν is the outpointing Euclidean unit normal;and
• αE is the volume (n− 1)-form induced by theEuclidean metric.
Bartnik/Chrusciel (1986): With weak fall-offconditions, the mass is well-defined & coordinateindependent.
136
Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
• ν is the outpointing Euclidean unit normal;and
• αE is the volume (n− 1)-form induced by theEuclidean metric.
Bartnik/Chrusciel (1986): With weak fall-offconditions, the mass is well-defined & coordinateindependent.
137
Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
• ν is the outpointing Euclidean unit normal;and
• αE is the volume (n− 1)-form induced by theEuclidean metric.
gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), s ∈ L1
138
Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
• ν is the outpointing Euclidean unit normal;and
• αE is the volume (n− 1)-form induced by theEuclidean metric.
Bartnik/Chrusciel (1986): With weak fall-offconditions, the mass is well-defined & coordinateindependent.
139
Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
• ν is the outpointing Euclidean unit normal;and
• αE is the volume (n− 1)-form induced by theEuclidean metric.
Bartnik/Chrusciel (1986): With weak fall-offconditions, the mass is well-defined & coordinateindependent.
140
Motivation:
141
Motivation:
When n = 3, ADM mass in general relativity.
142
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
143
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
144
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
145
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
146
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
g = −(
1− 2m%n−2
)dt2+
(1− 2m
%n−2
)−1
d%2+%2hSn−1
147
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
148
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
Two such regions fit together to formthe wormhole metric. Scalar-flat,AE, two ends. Not Ricci-flat, butconformally flat. Same mass m atboth ends: “size of throat.”
149
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
Two such regions fit together to formthe wormhole metric. Scalar-flat,AE, two ends. Not Ricci-flat, butconformally flat. Same mass m atboth ends: “size of throat.”
150
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
Two such regions fit together to formthe wormhole metric. Scalar-flat,AE, two ends. Not Ricci-flat, butconformally flat. Same mass m atboth ends: “size of throat.”
151
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
Two such regions fit together to formthe wormhole metric. Scalar-flat,AE, two ends. Not Ricci-flat, butconformally flat. Same mass m atboth ends: “size of throat.”
152
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
Two such regions fit together to formthe wormhole metric. Scalar-flat,AE, two ends. Not Ricci-flat, butconformally flat. Same mass m atboth ends: “size of throat.”
153
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
154
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
But any Ricci-flat ALE manifold has mass zero.
155
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
But any Ricci-flat ALE manifold has mass zero.
Bartnik: Ricci-flat =⇒ faster fall-off of metric!
156
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
But any Ricci-flat ALE manifold has mass zero.
Bartnik: Ricci-flat =⇒ faster fall-off of metric!
=⇒ “gravitational instantons” have mass zero.
157
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
Scalar-flat-Kahler Burns metric on C2⊂ C2 × CP1
158
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
Scalar-flat-Kahler Burns metric on C2 ⊂ C2×CP1
159
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
160
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0 hy-persurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
Scalar-flat-Kahler Burns metric on C2 ⊂ C2×CP1
161
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0 hy-persurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
Scalar-flat-Kahler Burns metric on C2 ⊂ C2×CP1
162
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0 hy-persurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
Scalar-flat-Kahler Burns metric on C2 ⊂ C2×CP1:
ω =i
2∂∂ [u + 3m log u] , u = |z1|2 + |z2|2
163
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0 hy-persurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
Scalar-flat-Kahler Burns metric on C2 ⊂ C2×CP1:
ω =i
2∂∂ [u + 3m log u] , u = |z1|2 + |z2|2
164
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0 hy-persurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
Scalar-flat-Kahler Burns metric on C2 ⊂ C2×CP1:
ω =i
2∂∂ [u + 3m log u] , u = |z1|2 + |z2|2
also has mass m .
165
Positive Mass Conjecture:
166
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
167
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
168
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
169
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Physical intuition:
170
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Physical intuition:
Local matter density ≥ 0 =⇒ total mass ≥ 0.
171
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Physical intuition:
Local matter density ≥ 0 =⇒ total mass ≥ 0.
172
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
173
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Schoen-Yau 1979:
174
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Schoen-Yau 1979:
Proved in dimension n ≤ 7.
175
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Schoen-Yau 1979:
Proved in dimension n ≤ 7.
Witten 1981:
176
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Schoen-Yau 1979:
Proved in dimension n ≤ 7.
Witten 1981:
Proved for spin manifolds (implicitly, for any n.)
177
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Schoen-Yau 1979:
Proved in dimension n ≤ 7.
Witten 1981:
Proved for spin manifolds (implicitly, for any n).
178
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Schoen-Yau 1979:
Proved in dimension n ≤ 7.
Witten 1981:
Proved for spin manifolds (implicitly, for any n).
Hawking-Pope 1978:
179
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Schoen-Yau 1979:
Proved in dimension n ≤ 7.
Witten 1981:
Proved for spin manifolds (implicitly, for any n).
Hawking-Pope 1978:
Conjectured true in ALE case, too.
180
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Schoen-Yau 1979:
Proved in dimension n ≤ 7.
Witten 1981:
Proved for spin manifolds (implicitly, for any n).
Hawking-Pope 1978:
Conjectured true in ALE case, too.
L 1987:
181
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Schoen-Yau 1979:
Proved in dimension n ≤ 7.
Witten 1981:
Proved for spin manifolds (implicitly, for any n).
Hawking-Pope 1978:
Conjectured true in ALE case, too.
L 1987:
ALE counter-examples.
182
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Schoen-Yau 1979:
Proved in dimension n ≤ 7.
Witten 1981:
Proved for spin manifolds (implicitly, for any n).
Hawking-Pope 1978:
Conjectured true in ALE case, too.
L 1987:
ALE counter-examples.
Scalar-flat Kahler metrics
183
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Schoen-Yau 1979:
Proved in dimension n ≤ 7.
Witten 1981:
Proved for spin manifolds (implicitly, for any n).
Hawking-Pope 1978:
Conjectured true in ALE case, too.
L 1987:
ALE counter-examples.
Scalar-flat Kahler metrics
on line bundles L→ CP1 of Chern-class ≤ −3.
184
Mass of ALE Kahler manifolds?
185
Mass of ALE Kahler manifolds?
Scalar-flat Kahler case?
186
Mass of ALE Kahler manifolds?
Scalar-flat Kahler case?
Lemma. Any ALE Kahler manifold has onlyone end.
187
Mass of ALE Kahler manifolds?
Scalar-flat Kahler case?
Lemma. Any ALE Kahler manifold has onlyone end.
188
Mass of ALE Kahler manifolds?
Scalar-flat Kahler case?
Lemma. Any ALE Kahler manifold has onlyone end.
189
Mass of ALE Kahler manifolds?
Scalar-flat Kahler case?
Lemma. Any ALE Kahler manifold has onlyone end.
...............................................................................................................................................................................................................................................................................................................................
...............................................................................................................................................................................................................................................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................. .....................................................
..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
..................................
..................................................
.....................
...............................
190
Mass of ALE Kahler manifolds?
Scalar-flat Kahler case?
Lemma. Any ALE Kahler manifold has onlyone end.
...............................................................................................................................................................................................................................................................................................................................
...............................................................................................................................................................................................................................................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................. .....................................................
..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
..................................
..................................................
.....................
...............................
n = 2m ≥ 4
191
Mass of ALE Kahler manifolds?
Scalar-flat Kahler case?
Lemma. Any ALE Kahler manifold has onlyone end.
192
Mass of ALE Kahler manifolds?
Scalar-flat Kahler case?
Lemma. Any ALE Kahler manifold has onlyone end.
Main Point:
193
Mass of ALE Kahler manifolds?
Scalar-flat Kahler case?
Lemma. Any ALE Kahler manifold has onlyone end.
Main Point:
Mass of an ALE Kahler manifold is unambiguous.
194
Mass of ALE Kahler manifolds?
Scalar-flat Kahler case?
Lemma. Any ALE Kahler manifold has onlyone end.
Main Point:
Mass of an ALE Kahler manifold is unambiguous.
Does not depend on the choice of an end!
195
We begin with the scalar-flat Kahler case.
196
We begin with the scalar-flat Kahler case.
Theorem A. The mass of an ALE scalar-flatKahler manifold is a topological invariant.
197
We begin with the scalar-flat Kahler case.
Theorem A. The mass of an ALE scalar-flatKahler manifold is a topological invariant.
198
We begin with the scalar-flat Kahler case.
Theorem A. The mass of an ALE scalar-flatKahler manifold is a topological invariant.
199
We begin with the scalar-flat Kahler case.
Theorem A. The mass of an ALE scalar-flatKahler manifold is a topological invariant.
200
We begin with the scalar-flat Kahler case.
Theorem A. The mass of an ALE scalar-flatKahler manifold is a topological invariant.
201
We begin with the scalar-flat Kahler case.
Theorem A. The mass of an ALE scalar-flatKahler manifold is a topological invariant.
That is, m(M, g, J) is completely determined by
202
We begin with the scalar-flat Kahler case.
Theorem A. The mass of an ALE scalar-flatKahler manifold is a topological invariant.
That is, m(M, g, J) is completely determined by
203
We begin with the scalar-flat Kahler case.
Theorem A. The mass of an ALE scalar-flatKahler manifold is a topological invariant.
That is, m(M, g, J) is completely determined by
• the smooth manifold M ,
204
We begin with the scalar-flat Kahler case.
Theorem A. The mass of an ALE scalar-flatKahler manifold is a topological invariant.
That is, m(M, g, J) is completely determined by
• the smooth manifold M ,
• the first Chern class c1 = c1(M,J) ∈ H2(M)of the complex structure, and
205
We begin with the scalar-flat Kahler case.
Theorem A. The mass of an ALE scalar-flatKahler manifold is a topological invariant.
That is, m(M, g, J) is completely determined by
• the smooth manifold M ,
• the first Chern class c1 = c1(M,J) ∈ H2(M)of the complex structure, and
• the Kahler class [ω] ∈ H2(M) of the metric.
206
We begin with the scalar-flat Kahler case.
Theorem A. The mass of an ALE scalar-flatKahler manifold is a topological invariant.
That is, m(M, g, J) is completely determined by
• the smooth manifold M ,
• the first Chern class c1 = c1(M,J) ∈ H2(M)of the complex structure, and
• the Kahler class [ω] ∈ H2(M) of the metric.
In fact, we will see that there is an explicit formulafor the mass in terms of these data!
207
The explicit formula reproduces the mass in caseswhere it previously had been laboriously computedfrom the definition. But it also allows one to quicklyread it off quite generally.
208
The explicit formula reproduces the mass in caseswhere it previously had been laboriously computedfrom the definition. But it also allows one to quicklyread it off quite generally.
209
The explicit formula reproduces the mass in caseswhere it previously had been laboriously computedfrom the definition. But it also allows one to quicklyread it off quite generally.
Corollary, suggested by Cristiano Spotti:
210
The explicit formula reproduces the mass in caseswhere it previously had been laboriously computedfrom the definition. But it also allows one to quicklyread it off quite generally.
Corollary, suggested by Cristiano Spotti:
Theorem B. Let (M4, g, J) be an ALE scalar-flat Kahler surface, and suppose that (M, g) isthe minimal resolution of a surface singularity.Then m(M, g) ≤ 0, with = iff g is Ricci-flat.
211
The explicit formula reproduces the mass in caseswhere it previously had been laboriously computedfrom the definition. But it also allows one to quicklyread it off quite generally.
Corollary, suggested by Cristiano Spotti:
Theorem B. Let (M4, g, J) be an ALE scalar-flat Kahler surface, and suppose that (M, g) isthe minimal resolution of a surface singularity.Then m(M, g) ≤ 0, with = iff g is Ricci-flat.
212
The explicit formula reproduces the mass in caseswhere it previously had been laboriously computedfrom the definition. But it also allows one to quicklyread it off quite generally.
Corollary, suggested by Cristiano Spotti:
Theorem B. Let (M4, g, J) be an ALE scalar-flat Kahler surface, and suppose that (M,J) isthe minimal resolution of a surface singularity.Then m(M, g) ≤ 0, with = iff g is Ricci-flat.
213
The explicit formula reproduces the mass in caseswhere it previously had been laboriously computedfrom the definition. But it also allows one to quicklyread it off quite generally.
Corollary, suggested by Cristiano Spotti:
Theorem B. Let (M4, g, J) be an ALE scalar-flat Kahler surface, and suppose that (M,J) isthe minimal resolution of a surface singularity.Then m(M, g) ≤ 0, with = iff g is Ricci-flat.
214
The explicit formula reproduces the mass in caseswhere it previously had been laboriously computedfrom the definition. But it also allows one to quicklyread it off quite generally.
Corollary, suggested by Cristiano Spotti:
Theorem B. Let (M4, g, J) be an ALE scalar-flat Kahler surface, and suppose that (M,J) isthe minimal resolution of a surface singularity.Then m(M, g) ≤ 0, with = iff g is Ricci-flat.
215
The explicit formula reproduces the mass in caseswhere it previously had been laboriously computedfrom the definition. But it also allows one to quicklyread it off quite generally.
Corollary, suggested by Cristiano Spotti:
Theorem B. Let (M4, g, J) be an ALE scalar-flat Kahler surface, and suppose that (M,J) isthe minimal resolution of a surface singularity.Then m(M, g) ≤ 0, with = iff g is Ricci-flat.
Note that minimality is essential here.
216
The explicit formula reproduces the mass in caseswhere it previously had been laboriously computedfrom the definition. But it also allows one to quicklyread it off quite generally.
Corollary, suggested by Cristiano Spotti:
Theorem B. Let (M4, g, J) be an ALE scalar-flat Kahler surface, and suppose that (M,J) isthe minimal resolution of a surface singularity.Then m(M, g) ≤ 0, with = iff g is Ricci-flat.
Note that minimality is essential here.
Non-minimal resolutions typically admit families ofsuch metrics for which the mass can be continuouslydeformed from negative to positive.
217
Explicit formula depends on a topological fact:
218
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. x Then the natural map
219
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. x Then the natural map
220
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
221
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
222
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
223
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
224
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
Here
Hpc (M) :=
ker d : Epc (M)→ Ep+1c (M)
dEp−1c (M)
225
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
Here
Hpc (M) :=
ker d : Epc (M)→ Ep+1c (M)
dEp−1c (M)
where
Epc (M) := Smooth, compactly supported p-forms on M.
226
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
227
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
Definition. If (M, g, J) is any ALE Kahler man-ifold, we will use
228
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
Definition. If (M, g, J) is any ALE Kahler man-ifold, we will use
229
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
Definition. If (M, g, J) is any ALE Kahler man-ifold, we will use
230
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
Definition. If (M, g, J) is any ALE Kahler man-ifold, we will use
♣ : H2dR(M)→ H2
c (M)
231
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
Definition. If (M, g, J) is any ALE Kahler man-ifold, we will use
♣ : H2dR(M)→ H2
c (M)
to denote the inverse of the natural map
232
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
Definition. If (M, g, J) is any ALE Kahler man-ifold, we will use
♣ : H2dR(M)→ H2
c (M)
to denote the inverse of the natural map
H2c (M)→ H2
dR(M)
induced by the inclusion of compactly supportedsmooth forms into all forms.
233
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
Definition. If (M, g, J) is any ALE Kahler man-ifold, we will use
♣ : H2dR(M)→ H2
c (M)
to denote the inverse of the natural map
H2c (M)→ H2
dR(M)
induced by the inclusion of compactly supportedsmooth forms into all forms.
234
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
235
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)ofcomplex dimension m has mass given by
236
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
237
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
238
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
239
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
240
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
241
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
242
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
243
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
where
244
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
where
• s = scalar curvature;
245
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
where
• s = scalar curvature;
• dµ = metric volume form;
246
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
where
• s = scalar curvature;
• dµ = metric volume form;
• c1 = c1(M,J) ∈ H2(M) is first Chern class;
247
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
where
• s = scalar curvature;
• dµ = metric volume form;
• c1 = c1(M,J) ∈ H2(M) is first Chern class;
• [ω] ∈ H2(M) is Kahler class of (g, J); and
248
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
where
• s = scalar curvature;
• dµ = metric volume form;
• c1 = c1(M,J) ∈ H2(M) is first Chern class;
• [ω] ∈ H2(M) is Kahler class of (g, J); and
• 〈 , 〉 is pairing between H2c (M) and H2m−2(M).
249
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
250
4πm(2m−1)(m−1)!
m(M, g) = − 4π(m−1)!
〈♣(c1), [ω]m−1〉+∫Msgdµg
251
For a compact Kahler manifold (M2m, g, J),
∫Msgdµg =
4π
(m− 1)!〈c1, [ω]m−1〉
252
For a compact Kahler manifold (M2m, g, J),
0 = − 4π
(m− 1)!〈c1, [ω]m−1〉 +
∫Msgdµg
253
For an ALE Kahler manifold (M2m, g, J),
4πm(2m−1)(m−1)!
m(M, g) = − 4π(m−1)!
〈♣(c1), [ω]m−1〉+∫Msgdµg
254
For an ALE Kahler manifold (M2m, g, J),
4πm(2m−1)(m−1)!
m(M, g) = − 4π(m−1)!
〈♣(c1), [ω]m−1〉+∫Msgdµg
So the mass is a “boundary correction” to the topo-logical formula for the total scalar curvature.
255
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
256
We can now state our mass formula:
Corollary. Any ALE scalar-flat Kahler mani-fold (M, g, J) of complex dimension m has massgiven by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
.
257
We can now state our mass formula:
Corollary. Any ALE scalar-flat Kahler mani-fold (M, g, J) of complex dimension m has massgiven by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
.
So Theorem A is an immediate consequence!
258
Rough Idea of Proof:
259
Rough Idea of Proof:
Special Case: Suppose
260
Rough Idea of Proof:
Special Case: Suppose
•m = 2, n = 4;
261
Rough Idea of Proof:
Special Case: Suppose
•m = 2, n = 4;
• Scalar flat: s ≡ 0; and
262
Rough Idea of Proof:
Special Case: Suppose
•m = 2, n = 4;
• Scalar flat: s ≡ 0; and
• Complex structure J standard at infinity:
263
Rough Idea of Proof:
Special Case: Suppose
•m = 2, n = 4;
• Scalar flat: s ≡ 0; and
• Complex structure J standard at infinity:
(M −K, J) ≈bih (C2 −B4)/Γ.
264
Rough Idea of Proof:
Special Case: Suppose
•m = 2, n = 4;
• Scalar flat: s ≡ 0; and
• Complex structure J standard at infinity:
(M −K, J) ≈bih (C2 −B4)/Γ.
Since g is Kahler, the complex coordinates
265
Rough Idea of Proof:
Special Case: Suppose
•m = 2, n = 4;
• Scalar flat: s ≡ 0; and
• Complex structure J standard at infinity:
(M −K, J) ≈bih (C2 −B4)/Γ.
Since g is Kahler, the complex coordinates
(z1, z2) = (x1 + ix2, x3 + ix4)
are harmonic. So xj are harmonic, too, and
266
Rough Idea of Proof:
Special Case: Suppose
•m = 2, n = 4;
• Scalar flat: s ≡ 0; and
• Complex structure J standard at infinity:
(M −K, J) ≈bih (C2 −B4)/Γ.
Since g is Kahler, the complex coordinates
(z1, z2) = (x1 + ix2, x3 + ix4)
are harmonic. So xj are harmonic, too, and
267
Rough Idea of Proof:
Special Case: Suppose
•m = 2, n = 4;
• Scalar flat: s ≡ 0; and
• Complex structure J standard at infinity:
(M −K, J) ≈bih (C2 −B4)/Γ.
Since g is Kahler, the complex coordinates
(z1, z2) = (x1 + ix2, x3 + ix4)
are harmonic. So xj are harmonic, too, and
gjk(gj`,k − gjk,`
)ν`αE = −?d log
(√det g
)+O(%−3−ε).
268
m(M, g) = − lim%→∞
1
12π2
∫S%/Γ
? d(
log√
det g)
269
m(M, g) = − lim%→∞
1
12π2
∫S%/Γ
? d(
log√
det g)
Now set θ = i2(∂ − ∂)
(log√
det g), so that
270
m(M, g) = − lim%→∞
1
12π2
∫S%/Γ
? d(
log√
det g)
Now set θ = i2(∂ − ∂)
(log√
det g), so that
ρ = dθ
is Ricci form, and
271
m(M, g) = − lim%→∞
1
12π2
∫S%/Γ
? d(
log√
det g)
Now set θ = i2(∂ − ∂)
(log√
det g), so that
ρ = dθ
is Ricci form, and
−?d log(√
det g)
= 2 θ ∧ ω.
272
m(M, g) = − lim%→∞
1
12π2
∫S%/Γ
? d(
log√
det g)
Now set θ = i2(∂ − ∂)
(log√
det g), so that
ρ = dθ
is Ricci form, and
−?d log(√
det g)
= 2 θ ∧ ω.Thus
m(M, g) = − lim%→∞
1
6π2
∫S%/Γ
θ ∧ ω
273
m(M, g) = − lim%→∞
1
12π2
∫S%/Γ
? d(
log√
det g)
Now set θ = i2(∂ − ∂)
(log√
det g), so that
ρ = dθ
is Ricci form, and
−?d log(√
det g)
= 2 θ ∧ ω.Thus
m(M, g) = − lim%→∞
1
6π2
∫S%/Γ
θ ∧ ω
However, since s = 0,
d(θ ∧ ω) = ρ ∧ ω =s
4ω2 = 0.
274
m(M, g) = − lim%→∞
1
12π2
∫S%/Γ
? d(
log√
det g)
Now set θ = i2(∂ − ∂)
(log√
det g), so that
ρ = dθ
is Ricci form, and
−?d log(√
det g)
= 2 θ ∧ ω.Thus
m(M, g) = − 1
6π2
∫S%/Γ
θ ∧ ω
275
Let f : M → R be smooth cut-off function:
276
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
......................
radius
1
.............................................................................................................................................................................................................................................................
...........................................................................................................................................................................................................................................................................
............................................................................................
...........................................................................................
277
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
......................
radius
1
.............................................................................................................................................................................................................................................................
...........................................................................................................................................................................................................................................................................
............................................................................................
...........................................................................................
278
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
......................
radius
1
.............................................................................................................................................................................................................................................................
...........................................................................................................................................................................................................................................................................
............................................................................................
...........................................................................................
279
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
......................
radius
“end”
1
.............................................................................................................................................................................................................................................................
...........................................................................................................................................................................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.
..........................................................................................................................................................................................................................................................................................................................................................................................................................................
............................................................................................
...........................................................................................
280
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
......................
radius
“end”
1
.............................................................................................................................................................................................................................................................
...........................................................................................................................................................................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.
..........................................................................................................................................................................................................................................................................................................................................................................................................................................
............................................................................................
...........................................................................................
...............................................................................................................................................................................................................................................................................................................................
...............................................................................................................................................................................................................................................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................. .....................................................
..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..............................................................
..................................
..................................
..................................................
.....................
...............................
281
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
......................
radius
1
.............................................................................................................................................................................................................................................................
...........................................................................................................................................................................................................................................................................
............................................................................................
...........................................................................................
282
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
......................
radius
M%
%
1
.............................................................................................................................................................................................................................................................
...........................................................................................................................................................................................................................................................................
.......
.......
.......
.......
.......
.......
.......
..
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
..........................................................................................................................................................................................................................................................................................................................................................................................................................................
.......
.......
.......
.....
............................................................................................
...........................................................................................
283
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
......................
radius
M%
%
1
.............................................................................................................................................................................................................................................................
...........................................................................................................................................................................................................................................................................
.......
.......
.......
.......
.......
.......
.......
..
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
..........................................................................................................................................................................................................................................................................................................................................................................................................................................
.......
.......
.......
.....
............................................................................................
...........................................................................................
.........................................................................
.........................................................................
......................................................................................................................................................................................................................................................
......................................................................................................................................................................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................. .....................................................
..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
..................................
..................................................
.....................
...............................
284
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
285
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
286
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
Compactly supported, because dθ = ρ near infinity.
287
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
288
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), [ω]〉 =
∫Mψ ∧ ω
289
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), [ω]〉 =
∫M%
ψ ∧ ω
where M% defined by radius ≤ %.
290
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), [ω]〉 =
∫M%
ψ ∧ ω
291
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), ω〉 =
∫M%
[ρ− d(fθ)] ∧ ω
292
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), ω〉 =
∫M%
[ρ− d(fθ)] ∧ ω
because scalar-flat =⇒ ρ ∧ ω = 0.
293
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), [ω]〉 = −∫M%
d(fθ) ∧ ω
because scalar-flat =⇒ ρ ∧ ω = 0.
294
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), [ω]〉 = −∫M%
d(fθ ∧ ω)
because scalar-flat =⇒ ρ ∧ ω = 0.
295
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), [ω]〉 = −∫M%
d(fθ ∧ ω)
296
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), [ω]〉 = −∫∂M%
fθ ∧ ω
by Stokes’ theorem.
297
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), [ω]〉 = −∫∂M%
θ ∧ ω
by Stokes’ theorem.
298
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), [ω]〉 = −∫S%/Γ
θ ∧ ω
by Stokes’ theorem.
299
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), [ω]〉 = −∫S%/Γ
θ ∧ ω
by Stokes’ theorem.
300
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), [ω]〉 = −∫S%/Γ
θ ∧ ω
by Stokes’ theorem.
So
m(M, g) = − 1
6π2
∫S%/Γ
θ ∧ ω
301
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), [ω]〉 = −∫S%/Γ
θ ∧ ω
by Stokes’ theorem.
So
m(M, g) = − 1
3π〈♣(c1), [ω]〉
302
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), [ω]〉 = −∫S%/Γ
θ ∧ ω
by Stokes’ theorem.
So
m(M, g) = − 1
3π〈♣(c1), [ω]〉
as claimed.
303
We assumed:
304
We assumed:
•m = 2;
• s ≡ 0; and
• Complex structure J standard at infinity.
305
General case:
306
General case:
•General m ≥ 2: straightforward. . .
307
General case:
•General m ≥ 2: straightforward. . .
308
General case:
•General m ≥ 2: straightforward. . .
• s 6≡ 0, compensate by adding∫s dµ. . .
309
General case:
•General m ≥ 2: straightforward. . .
• s 6≡ 0, compensate by adding∫s dµ. . .
• If m > 2, J is always standard at infinity.
310
General case:
•General m ≥ 2: straightforward. . .
• s 6≡ 0, compensate by adding∫s dµ. . .
• If m > 2, J is always standard at infinity.
• If m = 2 and AE, J is still standard at infinity.
311
General case:
•General m ≥ 2: straightforward. . .
• s 6≡ 0, compensate by adding∫s dµ. . .
• If m > 2, J is always standard at infinity.
• If m = 2 and AE, J is still standard at infinity.
• If m = 2 and ALE, J can be non-standard at∞.
312
General case:
•General m ≥ 2: straightforward. . .
• s 6≡ 0, compensate by adding∫s dµ. . .
• If m > 2, J is always standard at infinity.
• If m = 2 and AE, J is still standard at infinity.
• If m = 2 and ALE, J can be non-standard at∞.
The last point is serious.
313
General case:
•General m ≥ 2: straightforward. . .
• s 6≡ 0, compensate by adding∫s dµ. . .
• If m > 2, J is always standard at infinity.
• If m = 2 and AE, J is still standard at infinity.
• If m = 2 and ALE, J can be non-standard at∞.
Seen in “gravitational instantons”
and other explicit examples.
314
General case:
•General m ≥ 2: straightforward. . .
• s 6≡ 0, compensate by adding∫s dµ. . .
• If m > 2, J is always standard at infinity.
• If m = 2 and AE, J is still standard at infinity.
• If m = 2 and ALE, J can be non-standard at∞.
One argument proceeds by osculation:
315
General case:
•General m ≥ 2: straightforward. . .
• s 6≡ 0, compensate by adding∫s dµ. . .
• If m > 2, J is always standard at infinity.
• If m = 2 and AE, J is still standard at infinity.
• If m = 2 and ALE, J can be non-standard at∞.
One argument proceeds by osculation:
J = J0 + O(%−3), OJ = O(%−4)
in suitable asymptotic coordinates adapted to g.
316
To understand J at infinity:
317
To understand J at infinity:
Let M∞ be universal over of end M∞.
318
To understand J at infinity:
Let M∞ be universal over of end M∞.
Cap off M∞ by adding CPm−1 at infinity.
319
To understand J at infinity:
Let M∞ be universal over of end M∞.
Cap off M∞ by adding CPm−1 at infinity.
Added hypersurface CPm−1 has normal bundleO(1).
320
To understand J at infinity:
Let M∞ be universal over of end M∞.
Cap off M∞ by adding CPm−1 at infinity.
Added hypersurface CPm−1 has normal bundleO(1).
Complete analytic family encodes info about J .
321
To understand J at infinity:
322
To understand J at infinity:
AE case:
Compactify M itself by adding CPm−1 at infinity.
323
To understand J at infinity:
AE case:
Compactify M itself by adding CPm−1 at infinity.
Linear system of CPm−1 gives holomorphic map
(M ∪ CPm−1)→ CPmwhich is biholomorphism near CPm−1.
324
To understand J at infinity:
AE case:
Compactify M itself by adding CPm−1 at infinity.
Linear system of CPm−1 gives holomorphic map
(M ∪ CPm−1)→ CPmwhich is biholomorphism near CPm−1.
Thus obtain holomorphic map
Φ : M → Cm
which is biholomorphism near infinity.
325
To understand J at infinity:
AE case:
Compactify M itself by adding CPm−1 at infinity.
Linear system of CPm−1 gives holomorphic map
(M ∪ CPm−1)→ CPmwhich is biholomorphism near CPm−1.
Thus obtain holomorphic map
Φ : M → Cm
which is biholomorphism near infinity.
This has some interesting consequences. . .
326
Theorem D (Positive Mass Theorem). Any AEKahler manifold with non-negative scalar curva-ture has non-negative mass:
327
Theorem D (Positive Mass Theorem). Any AEKahler manifold with non-negative scalar curva-ture has non-negative mass:
328
Theorem D (Positive Mass Theorem). Any AEKahler manifold with non-negative scalar curva-ture has non-negative mass:
329
Theorem D (Positive Mass Theorem). Any AEKahler manifold with non-negative scalar curva-ture has non-negative mass:
330
Theorem D (Positive Mass Theorem). Any AEKahler manifold with non-negative scalar curva-ture has non-negative mass:
AE & Kahler & s ≥ 0 =⇒ m(M, g) ≥ 0.
331
Theorem D (Positive Mass Theorem). Any AEKahler manifold with non-negative scalar curva-ture has non-negative mass:
AE & Kahler & s ≥ 0 =⇒ m(M, g) ≥ 0.
Moreover, m = 0 ⇐⇒
332
Theorem D (Positive Mass Theorem). Any AEKahler manifold with non-negative scalar curva-ture has non-negative mass:
AE & Kahler & s ≥ 0 =⇒ m(M, g) ≥ 0.
Moreover, m = 0⇐⇒ (M, g) is Euclidean space.
333
Theorem D (Positive Mass Theorem). Any AEKahler manifold with non-negative scalar curva-ture has non-negative mass:
AE & Kahler & s ≥ 0 =⇒ m(M, g) ≥ 0.
Moreover, m = 0⇐⇒ (M, g) is Euclidean space.
Proof actually shows something stronger!
334
Theorem E (Penrose Inequality). Let (M, g, J)be an AE Kahler manifold with scalar curvatures ≥ 0. Then (M,J) carries a canonical divisorD that is expressed as a sum
∑jnjDj of com-
pact complex hypersurfaces with positive integercoefficients, with the property that
⋃jDj 6= ∅
whenever (M,J) 6= Cm. In terms of this divi-sor, we then have
335
Theorem E (Penrose Inequality). Let (M2m, g, J)be an AE Kahler manifold with scalar curvatures ≥ 0. Then (M,J) carries a canonical divisorD that is expressed as a sum
∑jnjDj of com-
pact complex hypersurfaces with positive integercoefficients, with the property that
⋃jDj 6= ∅
whenever (M,J) 6= Cm. In terms of this divi-sor, we then have
336
Theorem E (Penrose Inequality). Let (M2m, g, J)be an AE Kahler manifold with scalar curvatures ≥ 0. Then (M,J) carries a canonical divisorD that is expressed as a sum
∑jnjDj of com-
pact complex hypersurfaces with positive integercoefficients, with the property that
⋃jDj 6= ∅
whenever (M,J) 6= Cm. In terms of this divi-sor, we then have
337
Theorem E (Penrose Inequality). Let (M2m, g, J)be an AE Kahler manifold with scalar curvatures ≥ 0. Then (M,J) carries a canonical divisorD that is expressed as a sum
∑jnjDj of com-
pact complex hypersurfaces with positive integercoefficients, with the property that
⋃jDj 6= ∅
whenever (M,J) 6= Cm. In terms of this divi-sor, we then have
338
Theorem E (Penrose Inequality). Let (M2m, g, J)be an AE Kahler manifold with scalar curvatures ≥ 0. Then (M,J) carries a canonical divisorD that is expressed as a sum
∑jnjDj of com-
pact complex hypersurfaces with positive integercoefficients, with the property that
⋃jDj 6= ∅
whenever (M,J) 6= Cm. In terms of this divi-sor, we then have
339
Theorem E (Penrose Inequality). Let (M2m, g, J)be an AE Kahler manifold with scalar curvatures ≥ 0. Then (M,J) carries a canonical divisorD that is expressed as a sum
∑jnjDj of com-
pact complex hypersurfaces with positive integercoefficients, with the property that
⋃jDj 6= ∅
whenever (M,J) 6= Cm. In terms of this divi-sor, we then have
340
Theorem E (Penrose Inequality). Let (M2m, g, J)be an AE Kahler manifold with scalar curvatures ≥ 0. Then (M,J) carries a canonical divisorD that is expressed as a sum
∑jnjDj of com-
pact complex hypersurfaces with positive integercoefficients, with the property that
⋃jDj 6= ∅
whenever (M,J) 6= Cm. In terms of this divi-sor, we then have
341
Theorem E (Penrose Inequality). Let (M2m, g, J)be an AE Kahler manifold with scalar curvatures ≥ 0. Then (M,J) carries a canonical divisorD that is expressed as a sum
∑jnjDj of com-
pact complex hypersurfaces with positive integercoefficients, with the property that
⋃jDj 6= ∅
whenever (M,J) 6= Cm. In terms of this divi-sor, we then have
342
Theorem E (Penrose Inequality). Let (M2m, g, J)be an AE Kahler manifold with scalar curvatures ≥ 0. Then (M,J) carries a canonical divisorD that is expressed as a sum
∑jnjDj of com-
pact complex hypersurfaces with positive integercoefficients, with the property that
⋃jDj 6= ∅
whenever (M,J) 6= Cm. In terms of this divi-sor, we then have
343
Theorem E (Penrose Inequality). Let (M2m, g, J)be an AE Kahler manifold with scalar curvatures ≥ 0. Then (M,J) carries a canonical divisorD that is expressed as a sum
∑jnjDj of com-
pact complex hypersurfaces with positive integercoefficients, with the property that
⋃jDj 6= ∅
whenever (M,J) 6= Cm. In terms of this divi-sor, we then have
m(M, g) ≥ (m− 1)!
(2m− 1)πm−1
∑njVol (Dj)
344
Theorem E (Penrose Inequality). Let (M2m, g, J)be an AE Kahler manifold with scalar curvatures ≥ 0. Then (M,J) carries a canonical divisorD that is expressed as a sum
∑jnjDj of com-
pact complex hypersurfaces with positive integercoefficients, with the property that
⋃jDj 6= ∅
whenever (M,J) 6= Cm. In terms of this divi-sor, we then have
m(M, g) ≥ (m− 1)!
(2m− 1)πm−1
∑njVol (Dj)
345
Theorem E (Penrose Inequality). Let (M2m, g, J)be an AE Kahler manifold with scalar curvatures ≥ 0. Then (M,J) carries a canonical divisorD that is expressed as a sum
∑jnjDj of com-
pact complex hypersurfaces with positive integercoefficients, with the property that
⋃jDj 6= ∅
whenever (M,J) 6= Cm. In terms of this divi-sor, we then have
m(M, g) ≥ (m− 1)!
(2m− 1)πm−1
∑njVol (Dj)
346
Theorem E (Penrose Inequality). Let (M2m, g, J)be an AE Kahler manifold with scalar curvatures ≥ 0. Then (M,J) carries a canonical divisorD that is expressed as a sum
∑jnjDj of com-
pact complex hypersurfaces with positive integercoefficients, with the property that
⋃jDj 6= ∅
whenever (M,J) 6= Cm. In terms of this divi-sor, we then have
m(M, g) ≥ (m− 1)!
(2m− 1)πm−1
∑njVol (Dj)
with = ⇐⇒
347
Theorem E (Penrose Inequality). Let (M2m, g, J)be an AE Kahler manifold with scalar curvatures ≥ 0. Then (M,J) carries a canonical divisorD that is expressed as a sum
∑jnjDj of com-
pact complex hypersurfaces with positive integercoefficients, with the property that
⋃jDj 6= ∅
whenever (M,J) 6= Cm. In terms of this divi-sor, we then have
m(M, g) ≥ (m− 1)!
(2m− 1)πm−1
∑njVol (Dj)
with = ⇐⇒ (M, g, J) is scalar-flat Kahler.
348
This follows from existence of a holomorphic map
Φ : M → Cm
which is a biholomorphism near infinity.
349
This follows from existence of a holomorphic map
Φ : M → Cm
which is a biholomorphism near infinity.
Indeed, we then have a holomorphic section
ϕ = Φ∗dz1 ∧ · · · ∧ dzm
350
This follows from existence of a holomorphic map
Φ : M → Cm
which is a biholomorphism near infinity.
Indeed, we then have a holomorphic section
ϕ = Φ∗dz1 ∧ · · · ∧ dzm
of the canonical line bundle which vanishes exactlyat the critical points of Φ.
351
This follows from existence of a holomorphic map
Φ : M → Cm
which is a biholomorphism near infinity.
Indeed, we then have a holomorphic section
ϕ = Φ∗dz1 ∧ · · · ∧ dzm
of the canonical line bundle which vanishes exactlyat the critical points of Φ.
352
This follows from existence of a holomorphic map
Φ : M → Cm
which is a biholomorphism near infinity.
Indeed, we then have a holomorphic section
ϕ = Φ∗dz1 ∧ · · · ∧ dzm
of the canonical line bundle which vanishes exactlyat the critical points of Φ.
The zero set of ϕ, counted with multiplicities, givesus a canonical divisor
353
This follows from existence of a holomorphic map
Φ : M → Cm
which is a biholomorphism near infinity.
Indeed, we then have a holomorphic section
ϕ = Φ∗dz1 ∧ · · · ∧ dzm
of the canonical line bundle which vanishes exactlyat the critical points of Φ.
The zero set of ϕ, counted with multiplicities, givesus a canonical divisor
D =∑
njDj
354
This follows from existence of a holomorphic map
Φ : M → Cm
which is a biholomorphism near infinity.
Indeed, we then have a holomorphic section
ϕ = Φ∗dz1 ∧ · · · ∧ dzm
of the canonical line bundle which vanishes exactlyat the critical points of Φ.
The zero set of ϕ, counted with multiplicities, givesus a canonical divisor
D =∑
njDj
and
−〈♣(c1),ωm−1
(m− 1)!〉 =
∑njVol (Dj)
355
This follows from existence of a holomorphic map
Φ : M → Cm
which is a biholomorphism near infinity.
Indeed, we then have a holomorphic section
ϕ = Φ∗dz1 ∧ · · · ∧ dzm
of the canonical line bundle which vanishes exactlyat the critical points of Φ.
The zero set of ϕ, counted with multiplicities, givesus a canonical divisor
D =∑
njDj
and
−〈♣(c1),ωm−1
(m− 1)!〉 =
∑njVol (Dj)
so the mass formula implies the claim.
356
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
...............................................................................................................................................................................................................................................................................................................................
...............................................................................................................................................................................................................................................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................. .....................................................
..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
..................................
..................................................
.....................
...............................
357
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
358
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
...............................................................................................................................................................................................................................................................................................................................
...............................................................................................................................................................................................................................................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................. .....................................................
..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
..................................
..................................................
.....................
...............................
359
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
360
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
...............................................................................................................................................................................................................................................................................................................................
...............................................................................................................................................................................................................................................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................. .....................................................
..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
..................................
..................................................
.....................
...............................
361
362
Happy Birthday, Roger!
363
Happy Birthday, Roger!
Happy Birthday, Twistors!
364
Happy Birthday, Roger!
Happy Birthday, Twistors!
Happy Non-Retirement, Nick!
365