Anti-Self-Dual 4-Manifolds,
Quasi-Fuchsian Groups, &
Almost-Kahler Geometry
Claude LeBrunStony Brook University
Conformal and Symplectic GeometryUniversity of Auckland, 7 February, 2018
1
Most recent results joint with
Christopher J. BishopStony Brook University
e-print: arXiv:1708.03824 [math.DG]
4
Most recent results joint with
Christopher J. BishopStony Brook University
e-print: arXiv:1708.03824 [math.DG]
To appear in Comm. An. Geom.
5
Themes of this conference:
Conformal Geometry
Almost-Kahler Geometry
Symplectic GeometryJJJJJJJJJJJJJJJJJJJJ]
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6
Theme of this talk:
Conformal Geometry
Almost-Kahler Geometry
Symplectic GeometryJJJJJJJJJJ JJJJJJJJJJ]
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8
Theme of this talk:
Conformal Geometry
Almost-Kahler Geometry
Symplectic GeometryJJJJJJJJJJ JJJJJJJJJJ]
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Talk concerns phenomena specific to dimension 4.
9
Theme of this talk:
Conformal Geometry
Almost-Kahler Geometry
Symplectic GeometryJJJJJJJJJJ JJJJJJJJJJ]
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-�
Talk concerns phenomena specific to dimension 4.
Higher dimensions are demonstrably different.
10
Let (M2m, ω) be a compact symplectic manifold.
Thus, ω is a 2-form with dω = 0 and ω∧m 6= 0.
By convention, orient M so that ω∧m > 0.
15
Let (M2m, ω) be a compact symplectic manifold.
Thus, ω is a 2-form with dω = 0 and ω∧m 6= 0.
By convention, orient M so that ω∧m > 0.
⇒ ∃ compatible almost-complex structures J :
16
Let (M2m, ω) be a compact symplectic manifold.
Thus, ω is a 2-form with dω = 0 and ω∧m 6= 0.
By convention, orient M so that ω∧m > 0.
⇒ ∃ compatible almost-complex structures J :
J : TM → TM, J2 = −1,
17
Let (M2m, ω) be a compact symplectic manifold.
Thus, ω is a 2-form with dω = 0 and ω∧m 6= 0.
By convention, orient M so that ω∧m > 0.
⇒ ∃ compatible almost-complex structures J :
J : TM → TM, J2 = −1,
J∗ω = ω, ω(v, Jv) > 0 ∀v 6= 0.
18
Let (M2m, ω) be a compact symplectic manifold.
Thus, ω is a 2-form with dω = 0 and ω∧m 6= 0.
By convention, orient M so that ω∧m > 0.
⇒ ∃ compatible almost-complex structures J :
J : TM → TM, J2 = −1,
J∗ω = ω, ω(v, Jv) > 0 ∀v 6= 0.
19
Let (M2m, ω) be a compact symplectic manifold.
Thus, ω is a 2-form with dω = 0 and ω∧m 6= 0.
By convention, orient M so that ω∧m > 0.
⇒ ∃ compatible almost-complex structures J :
J : TM → TM, J2 = −1,
J∗ω = ω, ω(v, Jv) > 0 ∀v 6= 0.
Leads to theory of J -holomorphic curves,
20
Let (M2m, ω) be a compact symplectic manifold.
Thus, ω is a 2-form with dω = 0 and ω∧m 6= 0.
By convention, orient M so that ω∧m > 0.
⇒ ∃ compatible almost-complex structures J :
J : TM → TM, J2 = −1,
J∗ω = ω, ω(v, Jv) > 0 ∀v 6= 0.
Leads to theory of J -holomorphic curves,Gromov-Witten invariants, etc.
21
Let (M2m, ω) be a compact symplectic manifold.
Thus, ω is a 2-form with dω = 0 and ω∧m 6= 0.
By convention, orient M so that ω∧m > 0.
⇒ ∃ compatible almost-complex structures J :
J : TM → TM, J2 = −1,
J∗ω = ω, ω(v, Jv) > 0 ∀v 6= 0.
Leads to theory of J -holomorphic curves,Gromov-Witten invariants, etc.
Imitates Kahler geometry in a non-Kahler setting.
22
Let (M2m, ω) be a compact symplectic manifold.
Thus, ω is a 2-form with dω = 0 and ω∧m 6= 0.
By convention, orient M so that ω∧m > 0.
⇒ ∃ compatible almost-complex structures J :
J : TM → TM, J2 = −1,
J∗ω = ω, ω(v, Jv) > 0 ∀v 6= 0.
23
Let (M2m, ω) be a compact symplectic manifold.
Thus, ω is a 2-form with dω = 0 and ω∧m 6= 0.
By convention, orient M so that ω∧m > 0.
⇒ ∃ compatible almost-complex structures J :
J : TM → TM, J2 = −1,
J∗ω = ω, ω(v, Jv) > 0 ∀v 6= 0.
=⇒ g := ω(·, J ·) is a Riemannian metric.
24
Let (M2m, ω) be a compact symplectic manifold.
Thus, ω is a 2-form with dω = 0 and ω∧m 6= 0.
By convention, orient M so that ω∧m > 0.
⇒ ∃ compatible almost-complex structures J :
J : TM → TM, J2 = −1,
J∗ω = ω, ω(v, Jv) > 0 ∀v 6= 0.
=⇒ g := ω(·, J ·) is a Riemannian metric.
Such g are called almost-Kahler metrics, because
25
Let (M2m, ω) be a compact symplectic manifold.
Thus, ω is a 2-form with dω = 0 and ω∧m 6= 0.
By convention, orient M so that ω∧m > 0.
⇒ ∃ compatible almost-complex structures J :
J : TM → TM, J2 = −1,
J∗ω = ω, ω(v, Jv) > 0 ∀v 6= 0.
=⇒ g := ω(·, J ·) is a Riemannian metric.
Such g are called almost-Kahler metrics, because
Kahler ⇐⇒ J integrable.
26
Discussion involves three intertwined structures:
J
g
ωJJJJJJJJJJJJJJJJJJJJ]
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Any two algebraically determine the third.
29
Discussion involves three intertwined structures:
J
g
ωJJJJJJJJJJJJJJJJJJJJ]
��
-�
Any two algebraically determine the third.
For example, can avoid explicitly mentioning J .
30
Discussion involves three intertwined structures:
g
ω
��
Lemma.An oriented Riemannian manifold (M2m, g)is almost-Kahler w/ respect to the 2-form ω ⇐⇒
33
Discussion involves three intertwined structures:
g
ω
��
Lemma.An oriented Riemannian manifold (M2m, g)is almost-Kahler w/ respect to the 2-form ω ⇐⇒
34
Discussion involves three intertwined structures:
g
ω
��
Lemma.An oriented Riemannian manifold (M2m, g)is almost-Kahler w/ respect to the 2-form ω ⇐⇒
35
Discussion involves three intertwined structures:
g
ω
��
Lemma.An oriented Riemannian manifold (M2m, g)is almost-Kahler w/ respect to the 2-form ω ⇐⇒
36
Discussion involves three intertwined structures:
g
ω
��
Lemma.An oriented Riemannian manifold (M2m, g)is almost-Kahler w/ respect to the 2-form ω ⇐⇒• |ω|g ≡
√m,
37
Discussion involves three intertwined structures:
g
ω
��
Lemma.An oriented Riemannian manifold (M2m, g)is almost-Kahler w/ respect to the 2-form ω ⇐⇒• |ω|g ≡
√m,
• dω = 0, and
38
Discussion involves three intertwined structures:
g
ω
��
Lemma.An oriented Riemannian manifold (M2m, g)is almost-Kahler w/ respect to the 2-form ω ⇐⇒• |ω|g ≡
√m,
• dω = 0, and
• ∗ω =ω∧(m−1)
(m− 1)!.
39
Discussion involves three intertwined structures:
g
ω
��
Lemma.An oriented Riemannian manifold (M2m, g)is almost-Kahler w/ respect to the 2-form ω ⇐⇒• |ω|g ≡
√m,
• ω is a harmonic 2-form, and
• ∗ω =ω∧(m−1)
(m− 1)!.
40
Simplifies dramatically when m = 2:
g
ω
��
Lemma.An oriented Riemannian manifold (M2m, g)is almost-Kahler w/ respect to the 2-form ω ⇐⇒• |ω|g ≡
√m,
• ω is a harmonic 2-form, and
• ∗ω =ω∧(m−1)
(m− 1)!.
41
Simplifies dramatically when m = 2:
g
ω
��
Lemma.An oriented Riemannian 4-manifold (M, g)is almost-Kahler w/ respect to the 2-form ω ⇐⇒• |ω|g ≡
√2,
• ω is a harmonic 2-form, and
• ∗ω = ω.
42
On oriented (M4, g),
Λ2 = Λ+ ⊕ Λ−
Λ+ self-dual 2-forms:(+1)-eigenspace of ∗ : Λ2→ Λ2.
Λ− anti-self-dual 2-forms:(−1)-eigenspace of ∗ : Λ2→ Λ2.
45
Simplifies dramatically when m = 2:
g
ω
��
Lemma.An oriented Riemannian 4-manifold (M, g)is almost-Kahler w/ respect to the 2-form ω ⇐⇒• |ω|g ≡
√2,
• ω is a harmonic 2-form, and
• ∗ω = ω.
46
Simplifies dramatically when m = 2:
g
ω
��
Lemma.An oriented Riemannian 4-manifold (M, g)is almost-Kahler w/ respect to the 2-form ω ⇐⇒• |ω|g ≡
√2,
• ω is a self-dual harmonic 2-form.
47
On oriented (M4, g),
Λ2 = Λ+ ⊕ Λ−
Λ+ self-dual 2-forms:(+1)-eigenspace of ∗ : Λ2→ Λ2.
Λ− anti-self-dual 2-forms:(−1)-eigenspace of ∗ : Λ2→ Λ2.
48
On oriented (M4, g),
Λ2 = Λ+ ⊕ Λ−
Λ+ self-dual 2-forms:(+1)-eigenspace of ∗ : Λ2→ Λ2.
Λ− anti-self-dual 2-forms:(−1)-eigenspace of ∗ : Λ2→ Λ2.
But Hodge star
∗ : Λ2→ Λ2
is conformally invariant on middle-dimensional forms:
49
On oriented (M4, g),
Λ2 = Λ+ ⊕ Λ−
Λ+ self-dual 2-forms:(+1)-eigenspace of ∗ : Λ2→ Λ2.
Λ− anti-self-dual 2-forms:(−1)-eigenspace of ∗ : Λ2→ Λ2.
But Hodge star
∗ : Λ2→ Λ2
is conformally invariant on middle-dimensional forms:
Only depends on the conformal class
[g] := {u2g | u : M → R+}.
50
On oriented (M4, [g]),
Λ2 = Λ+ ⊕ Λ−
Λ+ self-dual 2-forms:(+1)-eigenspace of ∗ : Λ2→ Λ2.
Λ− anti-self-dual 2-forms:(−1)-eigenspace of ∗ : Λ2→ Λ2.
Only depends on the conformal class
[g] := {u2g | u : M → R+}.
51
Simplifies dramatically when m = 2:
g
ω
��
Lemma.An oriented Riemannian 4-manifold (M, g)is almost-Kahler w/ respect to the 2-form ω ⇐⇒• |ω|g ≡
√2,
• ω is a self-dual harmonic 2-form.
52
Simplifies dramatically when m = 2:
[g]
ω
��
Proposition.A conformal class [g] on a smoothcompact oriented 4-manifold M is represented byan almost-Kahler metric g iff it carries a self-dual harmonic 2-form ω that is 6= 0 everywhere.
53
Simplifies dramatically when m = 2:
[g]
ω
��
Proposition.A conformal class [g] on a smoothcompact oriented 4-manifold M is represented byan almost-Kahler metric g iff it carries a self-dual harmonic 2-form ω that is 6= 0 everywhere.
Moreover, the set of conformal classes [g] on Mthat carry such a harmonic form ω is open inthe C2 topology.
54
Hodge theory:
H2(M,R) = {ϕ ∈ Γ(Λ2) | dϕ = 0, d ∗ ϕ = 0}.Since ∗ is involution of RHS, =⇒
H2(M,R) = H+g ⊕H−g ,
56
Hodge theory:
H2(M,R) = {ϕ ∈ Γ(Λ2) | dϕ = 0, d ∗ ϕ = 0}.Since ∗ is involution of RHS, =⇒
H2(M,R) = H+g ⊕H−g ,
where
H±g = {ϕ ∈ Γ(Λ±) | dϕ = 0}self-dual & anti-self-dual harmonic forms.
57
Hodge theory:
H2(M,R) = {ϕ ∈ Γ(Λ2) | dϕ = 0, d ∗ ϕ = 0}.Since ∗ is involution of RHS, =⇒
H2(M,R) = H+g ⊕H−g ,
where
H±g = {ϕ ∈ Γ(Λ±) | dϕ = 0}self-dual & anti-self-dual harmonic forms.
One can choose a basis for H±g that depends
continuously on g in the C1,α topology.
58
Simplifies dramatically when m = 2:
[g]
ω
��
Proposition.A conformal class [g] on a smoothcompact oriented 4-manifold M is represented byan almost-Kahler metric g iff it carries a self-dual harmonic 2-form ω that is 6= 0 everywhere.
Moreover, the set of conformal classes [g] on Mthat carry such a harmonic form ω is open inthe C2 topology.
59
Simplifies dramatically when m = 2:“Conformal classes of symplectic type”
Proposition.A conformal class [g] on a smoothcompact oriented 4-manifold M is represented byan almost-Kahler metric g iff it carries a self-dual harmonic 2-form ω that is 6= 0 everywhere.
Moreover, the set of conformal classes [g] on Mthat carry such a harmonic form ω is open inthe C2 topology.
60
Hodge theory:
H2(M,R) = {ϕ ∈ Γ(Λ2) | dϕ = 0, d ∗ ϕ = 0}.Since ∗ is involution of RHS, =⇒
H2(M,R) = H+g ⊕H−g ,
where
H±g = {ϕ ∈ Γ(Λ±) | dϕ = 0}self-dual & anti-self-dual harmonic forms.
One can choose a basis for H±g that depends
continuously on g in the C1,α topology.
61
Hodge theory:
H2(M,R) = {ϕ ∈ Γ(Λ2) | dϕ = 0, d ∗ ϕ = 0}.Since ∗ is involution of RHS, =⇒
H2(M,R) = H+g ⊕H−g ,
where
H±g = {ϕ ∈ Γ(Λ±) | dϕ = 0}self-dual & anti-self-dual harmonic forms.
One can choose a basis for H±g that depends
continuously on g in the C1,α topology.
In particular, the numbers
b±(M) = dimH±gare independent of g, and so are invariants of M .
62
Best understood in terms of intersection pairing
H2(M,R)×H2(M,R) −→ R
( [ϕ] , [ψ] ) 7−→∫Mϕ ∧ ψ
Diagonalize:
+1. . .
+1︸ ︷︷ ︸b+(M)
b−(M)
−1. . .
−1
.
66
Best understood in terms of intersection pairing
H2(M,R)×H2(M,R) −→ R
( [ϕ] , [ψ] ) 7−→∫Mϕ ∧ ψ
Diagonalize:
+1. . .
+1︸ ︷︷ ︸b+(M)
b−(M)
−1. . .
−1
.
67
Best understood in terms of intersection pairing
H2(M,R)×H2(M,R) −→ R
( [ϕ] , [ψ] ) 7−→∫Mϕ ∧ ψ
Diagonalize:
+1. . .
+1︸ ︷︷ ︸b+(M)
b−(M)
−1. . .
−1
.
b2(M) = b+(M) + b−(M)
68
Best understood in terms of intersection pairing
H2(M,R)×H2(M,R) −→ R
( [ϕ] , [ψ] ) 7−→∫Mϕ ∧ ψ
Diagonalize:
+1. . .
+1︸ ︷︷ ︸b+(M)
b−(M)
−1. . .
−1
.
τ (M) = b+(M)− b−(M)
“Signature” of M .
69
Signature defined in terms of intersection pairing
H2(M,R)×H2(M,R) −→ R
( [ϕ] , [ψ] ) 7−→∫Mϕ ∧ ψ
Diagonalize:
+1. . .
+1︸ ︷︷ ︸b+(M)
b−(M)
−1. . .
−1
.
τ (M) = b+(M)− b−(M)
Signature of M .
70
Signature defined in terms of intersection pairing
H2(M,R)×H2(M,R) −→ R
( [ϕ] , [ψ] ) 7−→∫Mϕ ∧ ψ
τ (M) = b+(M)− b−(M)
71
Signature defined in terms of intersection pairing,
but also expressible as a curvature integral:
τ (M) =1
12π2
∫M
(|W+|2 − |W−|2
)dµ
= 〈p1(M), [M ]〉
72
Signature defined in terms of intersection pairing,
but also expressible as a curvature integral:
τ (M) =1
12π2
∫M
(|W+|2 − |W−|2
)dµ
= 〈13p1(M), [M ]〉
(Thom-Hirzebruch Signature Formula)
73
Signature defined in terms of intersection pairing,
but also expressible as a curvature integral:
τ (M) =1
12π2
∫M
(|W+|2 − |W−|2
)dµ
= 〈p1(M), [M ]〉
Has major consequences in conformal geometry.
74
Riemann curvature of g
R : Λ2→ Λ2
splits into 4 irreducible pieces:
Λ+∗ Λ−∗
Λ+ W+ + s12 r
Λ− r W− + s12
79
Riemann curvature of g
R : Λ2→ Λ2
splits into 4 irreducible pieces:
Λ+∗ Λ−∗
Λ+ W+ + s12 r
Λ− r W− + s12
where
s = scalar curvature
r = trace-free Ricci curvature
W+ = self-dual Weyl curvature (conformally invariant)
W− = anti-self-dual Weyl curvature
80
Riemann curvature of g
R : Λ2→ Λ2
splits into 4 irreducible pieces:
Λ+∗ Λ−∗
Λ+ W+ + s12 r
Λ− r W− + s12
where
s = scalar curvature
r = trace-free Ricci curvature
W+ = self-dual Weyl curvature (conformally invariant)
W− = anti-self-dual Weyl curvature ′′
81
For M4 compact, the Weyl functional
W ([g]) =
∫M
(|W+|2 + |W−|2
)dµg
measures the deviation from conformal flatness,
84
For M4 compact, the Weyl functional
W ([g]) =
∫M
(|W+|2 + |W−|2
)dµg
measures the deviation from conformal flatness,because (M4, g) is locally conformally flat ⇐⇒its Weyl curvature W = W+ + W− vanishes.
85
For M4 compact, the Weyl functional
W ([g]) =
∫M
(|W+|2 + |W−|2
)dµg
measures the deviation from conformal flatness,because (M4, g) is locally conformally flat ⇐⇒its Weyl curvature W = W+ + W− vanishes.
86
For M4 compact, the Weyl functional
W ([g]) =
∫M
(|W+|2 + |W−|2
)dµg
measures the deviation from conformal flatness,because (M4, g) is locally conformally flat ⇐⇒its Weyl curvature W = W+ + W− vanishes.
87
For M4 compact, the Weyl functional
W ([g]) =
∫M
(|W+|2 + |W−|2
)dµg
measures the deviation from conformal flatness,because (M4, g) is locally conformally flat ⇐⇒its Weyl curvature W = W+ + W− vanishes.
Basic problems: For given smooth compact M4,
88
For M4 compact, the Weyl functional
W ([g]) =
∫M
(|W+|2 + |W−|2
)dµg
measures the deviation from conformal flatness,because (M4, g) is locally conformally flat ⇐⇒its Weyl curvature W = W+ + W− vanishes.
Basic problems: For given smooth compact M4,
•What is inf W ?
89
For M4 compact, the Weyl functional
W ([g]) =
∫M
(|W+|2 + |W−|2
)dµg
measures the deviation from conformal flatness,because (M4, g) is locally conformally flat ⇐⇒its Weyl curvature W = W+ + W− vanishes.
Basic problems: For given smooth compact M4,
•What is inf W ?
•Do there exist minimizers?
90
For M4 compact, the Weyl functional
W ([g]) =
∫M
(|W+|2 + |W−|2
)dµg
measures the deviation from conformal flatness,because (M4, g) is locally conformally flat ⇐⇒its Weyl curvature W = W+ + W− vanishes.
But we’ve already noted that
12π2τ (M) =
∫M
(|W+|2 − |W−|2
)dµg
is a topological invariant.
91
For M4 compact, the Weyl functional
W ([g]) =
∫M
(|W+|2 + |W−|2
)dµg
measures the deviation from conformal flatness,because (M4, g) is locally conformally flat ⇐⇒its Weyl curvature W = W+ + W− vanishes.
But we’ve already noted that
12π2τ (M) =
∫M
(|W+|2 − |W−|2
)dµg
is a topological invariant.
So Weyl functional is essentially equivalent to
[g] 7−→∫M|W+|2dµg
92
For M4 compact, the Weyl functional
W ([g]) =
∫M
(|W+|2 + |W−|2
)dµg
measures the deviation from conformal flatness,because (M4, g) is locally conformally flat ⇐⇒its Weyl curvature W = W+ + W− vanishes.
But we’ve already noted that
12π2τ (M) =
∫M
(|W+|2 − |W−|2
)dµg
is a topological invariant.
In particular, metrics with W+ ≡ 0 minimize W .
93
For M4 compact, the Weyl functional
W ([g]) =
∫M
(|W+|2 + |W−|2
)dµg
measures the deviation from conformal flatness,because (M4, g) is locally conformally flat ⇐⇒its Weyl curvature W = W+ + W− vanishes.
But we’ve already noted that
12π2τ (M) =
∫M
(|W+|2 − |W−|2
)dµg
is a topological invariant.
In particular, metrics with W+ ≡ 0 minimize W .
If g has W+ ≡ 0, it is said to be anti-self-dual.
94
For M4 compact, the Weyl functional
W ([g]) =
∫M
(|W+|2 + |W−|2
)dµg
measures the deviation from conformal flatness,because (M4, g) is locally conformally flat ⇐⇒its Weyl curvature W = W+ + W− vanishes.
But we’ve already noted that
12π2τ (M) =
∫M
(|W+|2 − |W−|2
)dµg
is a topological invariant.
In particular, metrics with W+ ≡ 0 minimize W .
If g has W+ ≡ 0, it is said to be anti-self-dual.(ASD)
95
Twistor picture of anti-self-duality condition:
Oriented (M4, g)! (Z, J).
Z = S(Λ+), J : TZ → TZ, J2 = −1:
98
Twistor picture of anti-self-duality condition:
Oriented (M4, g)! (Z, J).
Z = S(Λ+), J : TZ → TZ, J2 = −1:
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r
rr S(Λ+)
M 4
↓
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99
Twistor picture of anti-self-duality condition:
Oriented (M4, g)! (Z, J).
Z = S(Λ+), J : TZ → TZ, J2 = −1:
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r
rr S(Λ+)
M 4
↓
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Theorem (Atiyah-Hitchin-Singer). (Z, J) is a com-plex 3-manifold iff W+ = 0.
100
Twistor picture of anti-self-duality condition:
Oriented (M4, g)! (Z, J).
Z = S(Λ+), J : TZ → TZ, J2 = −1:
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r
rr S(Λ+)
M 4
↓
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Theorem (Atiyah-Hitchin-Singer). (Z, J) is a com-plex 3-manifold iff W+ = 0.
Reconceptualizes earlier work by Penrose.
101
Twistor picture of anti-self-duality condition:
Oriented (M4, g)! (Z, J).
Z = S(Λ+), J : TZ → TZ, J2 = −1:
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r
rr S(Λ+)
M 4
↓
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Theorem (Atiyah-Hitchin-Singer). (Z, J) is a com-plex 3-manifold iff W+ = 0.
102
Twistor picture of anti-self-duality condition:
Oriented (M4, g)! (Z, J).
Z = S(Λ+), J : TZ → TZ, J2 = −1:
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r
rr S(Λ+)
M 4
↓
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Theorem (Atiyah-Hitchin-Singer). (Z, J) is a com-plex 3-manifold iff W+ = 0.
Motivates study of ASD metrics,and yields methods for constructing them.
103
A different link with complex geometry:
If (M4, g, J) is a Kahler surface, then [g] is ASD⇐⇒ the scalar curvature s of g is identically zero.
107
A different link with complex geometry:
If (M4, g, J) is a Kahler surface, then [g] is ASD⇐⇒ the scalar curvature s of g is identically zero.
Scalar-flat Kahler surfaces:
108
A different link with complex geometry:
If (M4, g, J) is a Kahler surface, then [g] is ASD⇐⇒ the scalar curvature s of g is identically zero.
Scalar-flat Kahler surfaces:special case of cscK manifolds,
109
A different link with complex geometry:
If (M4, g, J) is a Kahler surface, then [g] is ASD⇐⇒ the scalar curvature s of g is identically zero.
Scalar-flat Kahler surfaces:special case of cscK manifolds,and so of extremal Kahler manifolds.
110
A different link with complex geometry:
If (M4, g, J) is a Kahler surface, then [g] is ASD⇐⇒ the scalar curvature s of g is identically zero.
Scalar-flat Kahler surfaces:special case of cscK manifolds,and so of extremal Kahler manifolds.
Results proved about SFK in ’90s foreshadowedmany more recent results about general case.
111
A different link with complex geometry:
If (M4, g, J) is a Kahler surface, then [g] is ASD⇐⇒ the scalar curvature s of g is identically zero.
Scalar-flat Kahler surfaces:
112
A different link with complex geometry:
If (M4, g, J) is a Kahler surface, then [g] is ASD⇐⇒ the scalar curvature s of g is identically zero.
Scalar-flat Kahler surfaces:
Classification up to diffeomorphism: compact case
113
A different link with complex geometry:
If (M4, g, J) is a Kahler surface, then [g] is ASD⇐⇒ the scalar curvature s of g is identically zero.
Scalar-flat Kahler surfaces:
Classification up to diffeomorphism: (compact case)
114
A different link with complex geometry:
If (M4, g, J) is a Kahler surface, then [g] is ASD⇐⇒ the scalar curvature s of g is identically zero.
Scalar-flat Kahler surfaces:
Classification up to diffeomorphism: (compact case)
115
A different link with complex geometry:
If (M4, g, J) is a Kahler surface, then [g] is ASD⇐⇒ the scalar curvature s of g is identically zero.
Scalar-flat Kahler surfaces:
Classification up to diffeomorphism:
•Ricci-flat case (ignore from now on)
116
A different link with complex geometry:
If (M4, g, J) is a Kahler surface, then [g] is ASD⇐⇒ the scalar curvature s of g is identically zero.
Scalar-flat Kahler surfaces:
Classification up to diffeomorphism:
•Ricci-flat case (ignore from now on)
•Non-Ricci-flat case
117
A different link with complex geometry:
If (M4, g, J) is a Kahler surface, then [g] is ASD⇐⇒ the scalar curvature s of g is identically zero.
Scalar-flat Kahler surfaces:
Classification up to diffeomorphism:
•Ricci-flat case (ignore from now on)
–
–
–
•Non-Ricci-flat case
118
A different link with complex geometry:
If (M4, g, J) is a Kahler surface, then [g] is ASD⇐⇒ the scalar curvature s of g is identically zero.
Scalar-flat Kahler surfaces:
Classification up to diffeomorphism:
•Ricci-flat case (ignore from now on)
–K3
–
–
•Non-Ricci-flat case
119
A different link with complex geometry:
If (M4, g, J) is a Kahler surface, then [g] is ASD⇐⇒ the scalar curvature s of g is identically zero.
Scalar-flat Kahler surfaces:
Classification up to diffeomorphism:
•Ricci-flat case (ignore from now on)
–K3
– T 4
–
•Non-Ricci-flat case
120
A different link with complex geometry:
If (M4, g, J) is a Kahler surface, then [g] is ASD⇐⇒ the scalar curvature s of g is identically zero.
Scalar-flat Kahler surfaces:
Classification up to diffeomorphism:
•Ricci-flat case (ignore from now on)
–K3
– T 4
– eight specific finite quotients of these
•Non-Ricci-flat case
121
A different link with complex geometry:
If (M4, g, J) is a Kahler surface, then [g] is ASD⇐⇒ the scalar curvature s of g is identically zero.
Scalar-flat Kahler surfaces:
Classification up to diffeomorphism:
•Ricci-flat case (ignore from now on)
–K3
– T 4
– eight specific finite quotients of these
•Non-Ricci-flat case
122
A different link with complex geometry:
If (M4, g, J) is a Kahler surface, then [g] is ASD⇐⇒ the scalar curvature s of g is identically zero.
Scalar-flat Kahler surfaces:
Classification up to diffeomorphism:
•Ricci-flat case (ignore from now on)
•Non-Ricci-flat case
123
A different link with complex geometry:
If (M4, g, J) is a Kahler surface, then [g] is ASD⇐⇒ the scalar curvature s of g is identically zero.
Scalar-flat Kahler surfaces:
Classification up to diffeomorphism:
•Ricci-flat case (ignore from now on)
•Non-Ricci-flat case
–
–
–
–
124
A different link with complex geometry:
If (M4, g, J) is a Kahler surface, then [g] is ASD⇐⇒ the scalar curvature s of g is identically zero.
Scalar-flat Kahler surfaces:
Classification up to diffeomorphism:
•Ricci-flat case (ignore from now on)
•Non-Ricci-flat case
–CP2#kCP2, k ≥ 10
–
–
–
125
Convention:
CP2 = reverse oriented CP2.
Connected sum #:
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127
Convention:
CP2 = reverse oriented CP2.
Connected sum #:
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128
Convention:
CP2 = reverse oriented CP2.
Connected sum #:
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129
Convention:
CP2 = reverse oriented CP2.
Connected sum #:
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130
Convention:
CP2 = reverse oriented CP2.
Connected sum #:
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...
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131
Convention:
CP2 = reverse oriented CP2.
Connected sum #:
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132
Convention:
CP2 = reverse oriented CP2.
Connected sum #:
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133
Convention:
CP2 = reverse oriented CP2.
Connected sum #:
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134
Convention:
CP2 = reverse oriented CP2.
Connected sum #:
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Blowing Up: M M#CP2
135
Convention:
CP2 = reverse oriented CP2.
Connected sum #:
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136
Convention:
CP2 = reverse oriented CP2.
Connected sum #:
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Blowing Up: M M#CP2
137
Convention:
CP2 = reverse oriented CP2.
Connected sum #:
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138
A different link with complex geometry:
If (M4, g, J) is a Kahler surface, then [g] is ASD⇐⇒ the scalar curvature s of g is identically zero.
Scalar-flat Kahler surfaces:
Classification up to diffeomorphism:
•Ricci-flat case (ignore from now on)
•Non-Ricci-flat case
–CP2#kCP2, k ≥ 10
–
–
–
139
A different link with complex geometry:
If (M4, g, J) is a Kahler surface, then [g] is ASD⇐⇒ the scalar curvature s of g is identically zero.
Scalar-flat Kahler surfaces:
Classification up to diffeomorphism:
•Ricci-flat case (ignore from now on)
•Non-Ricci-flat case
–CP2#kCP2, k ≥ 10
– (T 2 × S2)#kCP2, k ≥ 1
–
–
140
A different link with complex geometry:
If (M4, g, J) is a Kahler surface, then [g] is ASD⇐⇒ the scalar curvature s of g is identically zero.
Scalar-flat Kahler surfaces:
Classification up to diffeomorphism:
•Ricci-flat case (ignore from now on)
•Non-Ricci-flat case
–CP2#kCP2, k ≥ 10
– (T 2 × S2)#kCP2, k ≥ 1
– Σ× S2
–
141
A different link with complex geometry:
If (M4, g, J) is a Kahler surface, then [g] is ASD⇐⇒ the scalar curvature s of g is identically zero.
Scalar-flat Kahler surfaces:
Classification up to diffeomorphism:
•Ricci-flat case (ignore from now on)
•Non-Ricci-flat case
–CP2#kCP2, k ≥ 10
– (T 2 × S2)#kCP2, k ≥ 1
– Σ× S2 and Σ×S2,
–
142
A different link with complex geometry:
If (M4, g, J) is a Kahler surface, then [g] is ASD⇐⇒ the scalar curvature s of g is identically zero.
Scalar-flat Kahler surfaces:
Classification up to diffeomorphism:
•Ricci-flat case (ignore from now on)
•Non-Ricci-flat case
–CP2#kCP2, k ≥ 10
– (T 2 × S2)#kCP2, k ≥ 1
– Σ× S2 and Σ×S2, genus Σ ≥ 2
–
143
A different link with complex geometry:
If (M4, g, J) is a Kahler surface, then [g] is ASD⇐⇒ the scalar curvature s of g is identically zero.
Scalar-flat Kahler surfaces:
Classification up to diffeomorphism:
•Ricci-flat case (ignore from now on)
•Non-Ricci-flat case
–CP2#kCP2, k ≥ 10
– (T 2 × S2)#kCP2, k ≥ 1
– Σ× S2 and Σ×S2, genus Σ ≥ 2
– (Σ× S2)#kCP2, k ≥ 1
144
Notice that the 4-manifolds CP2#kCP2do not admit scalar-flat Kahler metrics when k ≤ 9.
Plausible conjecture:
147
Notice that the 4-manifolds CP2#kCP2do not admit scalar-flat Kahler metrics when k ≤ 9.
Plausible conjecture:these manifolds don’t admit any ASD metrics.
148
Notice that the 4-manifolds CP2#kCP2do not admit scalar-flat Kahler metrics when k ≤ 9.
Plausible conjecture:these manifolds don’t admit any ASD metrics.
Stronger conjecture:
149
Notice that the 4-manifolds CP2#kCP2do not admit scalar-flat Kahler metrics when k ≤ 9.
Plausible conjecture:these manifolds don’t admit any ASD metrics.
Stronger conjecture:any metric on one of these manifolds M satisfies∫
M|W+|2dµ ≥
4π2
3(9− k)
150
Notice that the 4-manifolds CP2#kCP2do not admit scalar-flat Kahler metrics when k ≤ 9.
Plausible conjecture:these manifolds don’t admit any ASD metrics.
Stronger conjecture:any metric on one of these manifolds M satisfies∫
M|W+|2dµ ≥
4π2
3(2χ + 3τ )(M)
151
Notice that the 4-manifolds CP2#kCP2do not admit scalar-flat Kahler metrics when k ≤ 9.
Plausible conjecture:these manifolds don’t admit any ASD metrics.
Stronger conjecture:any metric on one of these manifolds M satisfies∫
M|W+|2dµ ≥
4π2
3(2χ + 3τ )(M)
Theorem (Gursky ’98).True for conformal classesof positive Yamabe constant.
152
Notice that the 4-manifolds CP2#kCP2do not admit scalar-flat Kahler metrics when k ≤ 9.
Plausible conjecture:these manifolds don’t admit any ASD metrics.
Stronger conjecture:any metric on one of these manifolds M satisfies∫
M|W+|2dµ ≥
4π2
3(2χ + 3τ )(M)
Theorem (Gursky ’98).True for conformal classesof positive Yamabe constant.
Theorem (L ’15). True for conformal classes ofsymplectic type.
153
Last result indicates that almost-Kahler conditiongives extra control on ASD conformal geometry.
154
Last result indicates that almost-Kahler conditiongives extra control on ASD conformal geometry.
Inyoung Kim ’16: classification of almost-KahlerASD roughly the same as in scalar-flat Kahler case.
155
Last result indicates that almost-Kahler conditiongives extra control on ASD conformal geometry.
Inyoung Kim ’16: classification of almost-KahlerASD roughly the same as in scalar-flat Kahler case.
Does this say anything about general ASD metrics?
156
Last result indicates that almost-Kahler conditiongives extra control on ASD conformal geometry.
Inyoung Kim ’16: classification of almost-KahlerASD roughly the same as in scalar-flat Kahler case.
Does this say anything about general ASD metrics?
Almost-Kahler ASD metrics sweep out an open setin the ASD moduli space.
157
Example.
Σ
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158
Example.
Σ
S2
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159
Example.
M = Σ× S2
Σ
S2
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160
Example.
M = Σ× S2
K = −1Σ
S2
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161
Example.
M = Σ× S2K = +1
K = −1Σ
S2
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162
Example.
M = Σ× S2K = +1
K = −1Σ
S2
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Product is scalar-flat Kahler.
163
Example.
M = Σ× S2K = +1
K = −1Σ
S2
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Product is scalar-flat Kahler.
164
Example.
M = Σ× S2K = +1
K = −1Σ
S2
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Product is scalar-flat Kahler.For both orientations!
165
Example.
M = Σ× S2K = +1
K = −1Σ
S2
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Product is scalar-flat Kahler.For both orientations!W+ = 0.
166
Example.
M = Σ× S2K = +1
K = −1Σ
S2
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Product is scalar-flat Kahler.For both orientations!W± = 0.
167
Example.
M = Σ× S2K = +1
K = −1Σ
S2
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Product is scalar-flat Kahler.For both orientations!W± = 0.Locally conformally flat!
168
Example.
M = Σ× S2K = +1
K = −1Σ
S2
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M = H2×S2= S4 − S1
169
Example.
M = Σ× S2K = +1
K = −1Σ
S2
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M = H2×S2 = S4 − S1
170
Example.
M = Σ× S2K = +1
K = −1Σ
S2
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M = H2×S2 = S4 − S1
π1(Σ) ↪→ SO+(1, 2)×SO(3) ↪→ SO+(1, 5)
171
Example.
M = Σ× S2K = +1
K = −1Σ
S2
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M = H2×S2 = S4 − S1
π1(Σ) ↪→ SO+(1, 2)× SO(3)↪→ SO+(1, 5)
172
Example.
M = Σ× S2K = +1
K = −1Σ
S2
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M = H2×S2 = S4 − S1
π1(Σ) ↪→ SO+(1, 2)× SO(3) ↪→ SO+(1, 5)
173
Example.
M = Σ× S2K = +1
K = −1Σ
S2
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Scalar-flat Kahler deformations: 12(g−1) moduli
174
Example.
M = Σ× S2K = +1
K = −1Σ
S2
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Scalar-flat Kahler deformations: 12(g−1) moduliLocally conformally flat def’ms: 30(g − 1) moduli
175
Example.
M = Σ× S2K = +1
K = −1Σ
S2
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Scalar-flat Kahler deformations: 12(g−1) modulialmost-Kahler ASD deformat’ns: 30(g−1) moduli
176
Last result indicates that almost-Kahler conditiongives extra control on ASD conformal geometry.
Inyoung Kim ’16: classification of almost-KahlerASD roughly the same as in scalar-flat Kahler case.
Does this say anything about general ASD metrics?
Almost-Kahler ASD metrics sweep out an open setin the ASD moduli space.
177
Last result indicates that almost-Kahler conditiongives extra control on ASD conformal geometry.
Inyoung Kim ’16: classification of almost-KahlerASD roughly the same as in scalar-flat Kahler case.
Does this say anything about general ASD metrics?
Almost-Kahler ASD metrics sweep out an open setin the ASD moduli space.
Is this subset also closed?
178
Last result indicates that almost-Kahler conditiongives extra control on ASD conformal geometry.
Inyoung Kim ’16: classification of almost-KahlerASD roughly the same as in scalar-flat Kahler case.
Does this say anything about general ASD metrics?
Almost-Kahler ASD metrics sweep out an open setin the ASD moduli space.
Is this subset also closed?
Does one get entire connected components this way?
179
Last result indicates that almost-Kahler conditiongives extra control on ASD conformal geometry.
Inyoung Kim ’16: classification of almost-KahlerASD roughly the same as in scalar-flat Kahler case.
Does this say anything about general ASD metrics?
Almost-Kahler ASD metrics sweep out an open setin the ASD moduli space.
Is this subset also closed?
Does one get entire connected components this way?
Alas, No!
180
Theorem A.Consider 4-manifolds M = Σ× S2,where Σ compact Riemann surface of genus g .
Then ∀ even g � 0, ∃ family [gt], t ∈ [0, 1], oflocally-conformally-flat classes on M , such that
185
Theorem A.Consider 4-manifolds M = Σ× S2,where Σ compact Riemann surface of genus g .
Then ∀ even g � 0, ∃ family [gt], t ∈ [0, 1], oflocally-conformally-flat classes on M , such that
186
Theorem A.Consider 4-manifolds M = Σ× S2,where Σ compact Riemann surface of genus g .
Then ∀ even g � 0, ∃ family [gt], t ∈ [0, 1], oflocally-conformally-flat classes on M , such that
187
Theorem A.Consider 4-manifolds M = Σ× S2,where Σ compact Riemann surface of genus g .
Then ∀ even g � 0, ∃ family [gt], t ∈ [0, 1], oflocally-conformally-flat classes on M , such that
188
Theorem A.Consider 4-manifolds M = Σ× S2,where Σ compact Riemann surface of genus g .
Then ∀ even g � 0, ∃ family [gt], t ∈ [0, 1], oflocally-conformally-flat classes on M , such that
189
Theorem A.Consider 4-manifolds M = Σ× S2,where Σ compact Riemann surface of genus g .
Then ∀ even g � 0, ∃ family [gt], t ∈ [0, 1], oflocally-conformally-flat classes on M , such that
190
Theorem A.Consider 4-manifolds M = Σ× S2,where Σ compact Riemann surface of genus g .
Then ∀ even g � 0, ∃ family [gt], t ∈ [0, 1], oflocally-conformally-flat classes on M , such that
• ∃ scalar-flat Kahler metric g0 ∈ [g0]; but
191
Theorem A.Consider 4-manifolds M = Σ× S2,where Σ compact Riemann surface of genus g .
Then ∀ even g � 0, ∃ family [gt], t ∈ [0, 1], oflocally-conformally-flat classes on M , such that
• ∃ scalar-flat Kahler metric g0 ∈ [g0]; but
• @ almost-Kahler metric g ∈ [g1].
192
Theorem A.Consider 4-manifolds M = Σ× S2,where Σ compact Riemann surface of genus g .
Then ∀ even g � 0, ∃ family [gt], t ∈ [0, 1], oflocally-conformally-flat classes on M , such that
• ∃ scalar-flat Kahler metric g0 ∈ [g0]; but
• @ almost-Kahler metric g ∈ [g1].
Same method simultaneously proves. . .
193
Theorem B. Fix an integer k ≥ 2, and thenconsider the 4-manifolds M = (Σ× S2)#kCP2,where Σ compact Riemann surface of genus g .
196
Theorem B. Fix an integer k ≥ 2, and thenconsider the 4-manifolds M = (Σ× S2)#kCP2,where Σ compact Riemann surface of genus g .
197
Theorem B. Fix an integer k ≥ 2, and thenconsider the 4-manifolds M = (Σ× S2)#kCP2,where Σ compact Riemann surface of genus g .
Then ∀ even g � 0, ∃ family [gt], t ∈ [0, 1], ofanti-self-dual conformal classes on M , such that
198
Theorem B. Fix an integer k ≥ 2, and thenconsider the 4-manifolds M = (Σ× S2)#kCP2,where Σ compact Riemann surface of genus g .
Then ∀ even g � 0, ∃ family [gt], t ∈ [0, 1], ofanti-self-dual conformal classes on M , such that
199
Theorem B. Fix an integer k ≥ 2, and thenconsider the 4-manifolds M = (Σ× S2)#kCP2,where Σ compact Riemann surface of genus g .
Then ∀ even g � 0, ∃ family [gt], t ∈ [0, 1], ofanti-self-dual conformal classes on M , such that
• ∃ scalar-flat Kahler metric g0 ∈ [g0]; but
200
Theorem B. Fix an integer k ≥ 2, and thenconsider the 4-manifolds M = (Σ× S2)#kCP2,where Σ compact Riemann surface of genus g .
Then ∀ even g � 0, ∃ family [gt], t ∈ [0, 1], ofanti-self-dual conformal classes on M , such that
• ∃ scalar-flat Kahler metric g0 ∈ [g0]; but
• @ almost-Kahler metric g ∈ [g1].
201
Theorem B. Fix an integer k ≥ 2, and thenconsider the 4-manifolds M = (Σ× S2)#kCP2,where Σ compact Riemann surface of genus g .
Then ∀ even g � 0, ∃ family [gt], t ∈ [0, 1], ofanti-self-dual conformal classes on M , such that
• ∃ scalar-flat Kahler metric g0 ∈ [g0]; but
• @ almost-Kahler metric g ∈ [g1].
Proof hinges on a construction of hyperbolic 3-manifolds.
202
Theorem B. Fix an integer k ≥ 2, and thenconsider the 4-manifolds M = (Σ× S2)#kCP2,where Σ compact Riemann surface of genus g .
Then ∀ even g � 0, ∃ family [gt], t ∈ [0, 1], ofanti-self-dual conformal classes on M , such that
• ∃ scalar-flat Kahler metric g0 ∈ [g0]; but
• @ almost-Kahler metric g ∈ [g1].
Proof hinges on a construction of hyperbolic 3-manifolds.
We begin by revisiting hyperbolic metrics on Σ.
203
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π1(Σ) ↪→ SO+(1, 2) = PSL(2,R)∩ ∩
SO+(1, 3) = PSL(2,C)
207
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π1(Σ) ↪→ SO+(1, 2) = PSL(2,R)∩ ∩
SO+(1, 3) = PSL(2,C)
208
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π1(Σ)∼=−→ Γ ⊂ PSL(2,R) Fuchsian group
211
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π1(Σ)∼=−→ Γ ⊂ PSL(2,C) Fuchsian group
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....................
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.....
...................................
... ................. ........................ ............... ...........
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......... .............
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....................
Fuchsian quasi-Fuchsian
215
.......
.......
.......
.......
.......
.......
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..........................................
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....................
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.....
...................................
... ................. ........................ ............... ...........
............... .......................... ................... .....
......... .............
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...............
....................
Fuchsian quasi-Fuchsian
216
.......
.......
.......
.......
.......
.......
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.....................................
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.....
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... ................. ........................ ............... ...........
............... .......................... ................... .....
......... .............
............... ................ ....................................... ..................... ................ ...
...............
....................
Fuchsian quasi-Fuchsian
π1(Σ)∼=−→ Γ ⊂ PSL(2,C) quasi-Fuchsian group
217
.......
.......
.......
.......
.......
.......
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....................
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.....
...................................
... ................. ........................ ............... ...........
............... .......................... ................... .....
......... .............
............... ................ ....................................... ..................... ................ ...
...............
....................
Fuchsian quasi-Fuchsian
π1(Σ)∼=−→ Γ ⊂ PSL(2,C) quasi-Fuchsian group
of Bers type
218
.......
.......
.......
.......
.......
.......
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....................
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.....
...................................
... ................. ........................ ............... ...........
............... .......................... ................... .....
......... .............
............... ................ ....................................... ..................... ................ ...
...............
....................
Fuchsian quasi-Fuchsian
π1(Σ)∼=−→ Γ ⊂ PSL(2,C) quasi-Fuchsian group
of Bers type
Quasi-conformally conjugate to Fuchsian.
219
H3
........................................................
........................................
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....................
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...................
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.......................... ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .............
..........................
220
.....................................................................................................................................................................................................................................................................................................................................................................................................
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................................................................................. .....................................................
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............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
221
X = H3/Γ
.....................................................................................................................................................................................................................................................................................................................................................................................................
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................................................................................. .....................................................
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................................................................................. .....................................................
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.................................................................
............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
222
X = H3/Γ
.....................................................................................................................................................................................................................................................................................................................................................................................................
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................................................................................. .....................................................
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................
................................................................................. .....................................................
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............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
Γ Fuchsian
223
X = H3/Γ
.....................................................................................................................................................................................................................................................................................................................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................
................
................................................................................. .....................................................
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.....................................................................................................................................................................................................................................................................................................................................................................................
................
................................................................................. .....................................................
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............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
Γ Fuchsian
X ≈ Σ× R
224
X = H3/Γ
....................................
....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.................................
.....................................................................................................................................................................................................................................................................................................................................................................................
................
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................
................
.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................
..................
..................
..................
..................
............
Γ quasi-Fuchsian
X ≈ Σ× R
225
X = H3/Γ
....................................
....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.................................
.....................................................................................................................................................................................................................................................................................................................................................................................
................
................................................................................. .....................................................
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................
................
.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................
..................
..................
..................
..................
............
Γ quasi-Fuchsian
X ≈ Σ× R
Freedom: two points in Teichmuller space.
226
X = H3/Γ
....................................
....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.................................
.....................................................................................................................................................................................................................................................................................................................................................................................
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.................................................................
................
................
.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................
..................
..................
..................
..................
............
Γ quasi-Fuchsian
X ≈ Σ× R
227
X = H3/Γ
....................................
....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.................................
.....................................................................................................................................................................................................................................................................................................................................................................................
................
................................................................................. .....................................................
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................
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..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................
..................
..................
..................
..................
............
Γ quasi-Fuchsian
X ≈ Σ× [0, 1]
228
s
s
1
0
f.......................................................................................................................................
...........................
X = H3/Γ
....................................
....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.................................
.....................................................................................................................................................................................................................................................................................................................................................................................
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................
.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................
..................
..................
..................
..................
............
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
Γ quasi-Fuchsian
X ≈ Σ× [0, 1]
Tunnel-Vision function:
f : X → [0, 1]
229
s
s
1
0
f.......................................................................................................................................
...........................
X = H3/Γ
....................................
....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
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................................................................................. .....................................................
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................
.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................
..................
..................
..................
..................
............
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
Γ quasi-Fuchsian
X ≈ Σ× [0, 1]
Tunnel-Vision function:
f : X → [0, 1]
∆f = 0
230
.......
.......
.......
.......
.......
.......
...................................................................................................................................................................................................................................................
.....................................
.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
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....................
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.......
.......
.......
.......
.......
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.......................................................................
.........................................
............................................... ...................................
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................... .............
............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ................... ......
....................
....................................................................... ................. ...
.....
...................................
... ................. ........................ ............... ...........
............... .......................... ................... .....
......... .............
............... ................ ....................................... ..................... ................ ...
...............
....................
Fuchsian quasi-Fuchsian
231
s
s
1
0
f.......................................................................................................................................
...........................
X = H3/Γ
....................................
....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.................................
.....................................................................................................................................................................................................................................................................................................................................................................................
................
................................................................................. .....................................................
......................................................................................................................................
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.......................................................................................................................................................................................................................................................................................................................................................................................................................................
................
................................................................................. .....................................................
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................
.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................
..................
..................
..................
..................
............
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
Γ quasi-Fuchsian
X ≈ Σ× [0, 1]
Tunnel-Vision function:
f : X → [0, 1]
∆f = 0
232
X = H3/Γ
....................................
....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
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................................................................................. .....................................................
......................................................................................................................................
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.......................................................................................................................................................................................................................................................................................................................................................................................................................................
................
................................................................................. .....................................................
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................
................
.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................
..................
..................
..................
..................
............
233
X = H3/Γ
....................................
....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.................................
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................................................................................. .....................................................
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.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................
..................
..................
..................
..................
............
Construction of conformally flat 4-manifolds:
234
X = H3/Γ
....................................
....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.................................
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................................................................................. .....................................................
......................................................................................................................................
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.......................................................................................................................................................................................................................................................................................................................................................................................................................................
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................................................................................. .....................................................
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.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................
..................
..................
..................
..................
............
Construction of conformally flat 4-manifolds:
M = [X × S1]/ ∼
235
X = H3/Γ
....................................
....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.................................
.....................................................................................................................................................................................................................................................................................................................................................................................
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................................................................................. .....................................................
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................................................................................. .....................................................
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.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................
..................
..................
..................
..................
............
Construction of conformally flat 4-manifolds:
M = [X × S1]/ ∼
∼: crush ∂X × S1 to ∂X .
236
X = H3/Γ
....................................
....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.................................
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Construction of conformally flat 4-manifolds:
M = [X × S1]/ ∼
237
X = H3/Γ
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Construction of conformally flat 4-manifolds:
M = [X × S1]/ ∼
g = f (1− f )[h + dt2]
238
X = H3/Γ
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Construction of conformally flat 4-manifolds:
M = [X × S1]/ ∼
g = f (1− f )[h + dt2]
239
X = H3/Γ
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Construction of conformally flat 4-manifolds:
M = [X × S1]/ ∼
g = f (1− f )[h + dt2]
Fuchsian case: Σ× S2 scalar-flat Kahler.
240
X = H3/Γ
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241
X = H3/Γ
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Construction of ASD 4-manifolds:
242
s ss
ssX = H3/Γ
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Construction of ASD 4-manifolds:
Choose k points p1, . . . , pk ∈ X
243
s
s
1
0
f.......................................................................................................................................
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s ss
ssX = H3/Γ
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Construction of ASD 4-manifolds:
Choose k points p1, . . . , pk ∈ X
satisfying∑kj=1 f (pj) ∈ Z.
244
s
s
1
0
f.......................................................................................................................................
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s ss
ssX = H3/Γ
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Construction of ASD 4-manifolds:
Choose k points p1, . . . , pk ∈ X
satisfying∑kj=1 f (pj) ∈ Z.
Can do if k 6= 1.
245
s ss
ssX = H3/Γ
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Construction of ASD 4-manifolds:
Let Gj be the Green’s function of pj:
246
s ss
ssX = H3/Γ
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Construction of ASD 4-manifolds:
Let Gj be the Green’s function of pj:
∆Gj = 2πδpj, Gj → 0 at ∂X
247
s ss
ssX = H3/Γ
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Construction of ASD 4-manifolds:
Let Gj be the Green’s function of pj, and set
V = 1 +
k∑j=1
Gj.
248
s ss
ssX = H3/Γ
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Construction of ASD 4-manifolds:
V = 1 +
k∑j=1
Gj.
249
s ss
ssX = H3/Γ
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Construction of ASD 4-manifolds:
V = 1 +
k∑j=1
Gj.
Choose P → (X−{p1, . . . , pk}) circle bundle withconnection form θ such that
dθ = ?dV .
250
s ss
ssX = H3/Γ
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Construction of ASD 4-manifolds:
g = f (1− f )[V h + V −1θ2]
V = 1 +
k∑j=1
Gj
dθ = ?dV
251
s ss
ssX = H3/Γ
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Construction of ASD 4-manifolds:
g = f (1− f )[V h + V −1θ2]
V = 1 +
k∑j=1
Gj
dθ = ?dV
252
s ss
ssX = H3/Γ
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Construction of ASD 4-manifolds:
g = f (1− f )[V h + V −1θ2]
M = P ∪ {p1, . . . , pk} ∪ ∂X
253
s ss
ssX = H3/Γ
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Construction of ASD 4-manifolds:
g = f (1− f )[V h + V −1θ2]
M = P ∪ {p1, . . . , pk} ∪ ∂X↓ ↓ ↓ ↓X = X − {p1, . . . , pk} ∪ {p1, . . . , pk} ∪ ∂X
254
s ss
ssX = H3/Γ
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Construction of ASD 4-manifolds:
g = f (1− f )[V h + V −1θ2]
M = P ∪ {p1, . . . , pk} ∪ ∂X
255
s ss
ssX = H3/Γ
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Construction of ASD 4-manifolds:
g = f (1− f )[V h + V −1θ2]
M = P ∪ {p1, . . . , pk} ∪ ∂X
≈(Σ× S2)#kCP2 scalar-flat Kahler
256
.......................................................................................................................
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s s
Σ
X = Σ× [0, 1]
.......
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257
s ss
ssX = H3/Γ
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Construction of ASD 4-manifolds:
g = f (1− f )[V h + V −1θ2]
M = P ∪ {p1, . . . , pk} ∪ ∂X
≈(Σ× S2)#kCP2 scalar-flat Kahler
258
s ss
ssX = H3/Γ
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Construction of ASD 4-manifolds:
g = f (1− f )[V h + V −1θ2]
M = P ∪ {p1, . . . , pk} ∪ ∂X
Fuchsian case: (Σ× S2)#kCP2 scalar-flat Kahler
259
s
s
1
0
f.......................................................................................................................................
...........................
X = H3/Γ
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Γ quasi-Fuchsian
X ≈ Σ× [0, 1]
Tunnel-Vision function:
f : X → [0, 1]
∆f = 0
260
Theorem.Let (M, [g]) be ASD manifold arisingfrom a quasi-Fuchsian 3-manifold (X, h) and aconfiguration of points in X.
Then ∃ almost-Kahler g ∈ [g] ⇐⇒ tunnel-visionfunction f : X → (0, 1) has no critical points.
261
Theorem.Let (M, [g]) be ASD manifold arisingfrom a quasi-Fuchsian 3-manifold (X, h) and aconfiguration of points in X.
Then ∃ almost-Kahler g ∈ [g] ⇐⇒ tunnel-visionfunction f : X → (0, 1) has no critical points.
262
Theorem.Let (M, [g]) be ASD manifold arisingfrom a quasi-Fuchsian 3-manifold (X, h) and aconfiguration of points in X.
Then ∃ almost-Kahler g ∈ [g] ⇐⇒ tunnel-visionfunction f : X → (0, 1) has no critical points.
263
Theorem.Let (M, [g]) be ASD manifold arisingfrom a quasi-Fuchsian 3-manifold (X, h) and aconfiguration of points in X.
Then ∃ almost-Kahler g ∈ [g] ⇐⇒ tunnel-visionfunction f : X → (0, 1) has no critical points.
264
Theorem.Let (M, [g]) be ASD manifold arisingfrom a quasi-Fuchsian 3-manifold (X, h) and aconfiguration of points in X.
Then ∃ almost-Kahler g ∈ [g] ⇐⇒ tunnel-visionfunction f : X → (0, 1) has no critical points.
265
Theorem.Let (M, [g]) be ASD manifold arisingfrom a quasi-Fuchsian 3-manifold (X, h) and aconfiguration of points in X.
Then ∃ almost-Kahler g ∈ [g] ⇐⇒ tunnel-visionfunction f : X → (0, 1) has no critical points.
266
Theorem.Let (M, [g]) be ASD manifold arisingfrom a quasi-Fuchsian 3-manifold (X, h) and aconfiguration of points in X.
Then ∃ almost-Kahler g ∈ [g] ⇐⇒ tunnel-visionfunction f : X → (0, 1) has no critical points.
267
Theorem.Let (M, [g]) be ASD manifold arisingfrom a quasi-Fuchsian 3-manifold (X, h) and aconfiguration of points in X.
Then ∃ almost-Kahler g ∈ [g] ⇐⇒ tunnel-visionfunction f : X → (0, 1) has no critical points.
268
Theorem.Let (M, [g]) be ASD manifold arisingfrom a quasi-Fuchsian 3-manifold (X, h) and aconfiguration of points in X.
Then ∃ almost-Kahler g ∈ [g] ⇐⇒ tunnel-visionfunction f : X → (0, 1) has no critical points.
Proof.
269
Theorem.Let (M, [g]) be ASD manifold arisingfrom a quasi-Fuchsian 3-manifold (X, h) and aconfiguration of points in X.
Then ∃ almost-Kahler g ∈ [g] ⇐⇒ tunnel-visionfunction f : X → (0, 1) has no critical points.
Proof.
b+[(Σ× S2)#kCP2] = 1.
270
Theorem.Let (M, [g]) be ASD manifold arisingfrom a quasi-Fuchsian 3-manifold (X, h) and aconfiguration of points in X.
Then ∃ almost-Kahler g ∈ [g] ⇐⇒ tunnel-visionfunction f : X → (0, 1) has no critical points.
Proof.
b+[(Σ× S2)#kCP2] = 1.
ω = df ∧ θ + V ? df .
271
Theorem.Let (M, [g]) be ASD manifold arisingfrom a quasi-Fuchsian 3-manifold (X, h) and aconfiguration of points in X.
Then ∃ almost-Kahler g ∈ [g] ⇐⇒ tunnel-visionfunction f : X → (0, 1) has no critical points.
Proof.
b+[(Σ× S2)#kCP2] = 1.
ω = df ∧ θ + V ? df .
272
Lemma.For any piecewise smooth Jordan curveγ ⊂ C and any ε > 0, there is a positive integerN such that, for every compact oriented surfaceΣ of genus g ≥ N , there is quasi-Fuchsian groupΓ ∼= π1(Σ) whose limit set Λ(Γ) ⊂ C ⊂ CP1 iswithin Hausdorff distance ε of γ.
If γ is invariant under ζ 7→ −ζ, and if g is even,we can also arrange for Λ(Γ) to also be invariantunder reflection through the origin.
273
Lemma.For any piecewise smooth Jordan curveγ ⊂ C and any ε > 0, there is a positive integerN such that, for every compact oriented surfaceΣ of genus g ≥ N , there is quasi-Fuchsian groupΓ ∼= π1(Σ) whose limit set Λ(Γ) ⊂ C ⊂ CP1 iswithin Hausdorff distance ε of γ.
If γ is invariant under ζ 7→ −ζ, and if g is even,we can also arrange for Λ(Γ) to also be invariantunder reflection through the origin.
274
Lemma.For any piecewise smooth Jordan curveγ ⊂ C and any ε > 0, there is a positive integerN such that, for every compact oriented surfaceΣ of genus g ≥ N , there is quasi-Fuchsian groupΓ ∼= π1(Σ) whose limit set Λ(Γ) ⊂ C ⊂ CP1 iswithin Hausdorff distance ε of γ.
If γ is invariant under ζ 7→ −ζ, and if g is even,we can also arrange for Λ(Γ) to also be invariantunder reflection through the origin.
275
Lemma.For any piecewise smooth Jordan curveγ ⊂ C and any ε > 0, there is a positive integerN such that, for every compact oriented surfaceΣ of genus g ≥ N , there is quasi-Fuchsian groupΓ ∼= π1(Σ) whose limit set Λ(Γ) ⊂ C ⊂ CP1 iswithin Hausdorff distance ε of γ.
If γ is invariant under ζ 7→ −ζ, and if g is even,we can also arrange for Λ(Γ) to also be invariantunder reflection through the origin.
276
Lemma.For any piecewise smooth Jordan curveγ ⊂ C and any ε > 0, there is a positive integerN such that, for every compact oriented surfaceΣ of genus g ≥ N , there is quasi-Fuchsian groupΓ ∼= π1(Σ) whose limit set Λ(Γ) ⊂ C ⊂ CP1 iswithin Hausdorff distance ε of γ.
If γ is invariant under ζ 7→ −ζ, and if g is even,we can also arrange for Λ(Γ) to also be invariantunder reflection through the origin.
277
Lemma.For any piecewise smooth Jordan curveγ ⊂ C and any ε > 0, there is a positive integerN such that, for every compact oriented surfaceΣ of genus g ≥ N , there is quasi-Fuchsian groupΓ ∼= π1(Σ) whose limit set Λ(Γ) ⊂ C ⊂ CP1 iswithin Hausdorff distance ε of γ.
If γ is invariant under ζ 7→ −ζ, and if g is even,we can also arrange for Λ(Γ) to also be invariantunder reflection through the origin.
278
Lemma.For any piecewise smooth Jordan curveγ ⊂ C and any ε > 0, there is a positive integerN such that, for every compact oriented surfaceΣ of genus g ≥ N , there is quasi-Fuchsian groupΓ ∼= π1(Σ) whose limit set Λ(Γ) ⊂ C ⊂ CP1 iswithin Hausdorff distance ε of γ.
If γ is invariant under ζ 7→ −ζ, and if g is even,we can also arrange for Λ(Γ) to also be invariantunder reflection through the origin.
279
Lemma.For any piecewise smooth Jordan curveγ ⊂ C and any ε > 0, there is a positive integerN such that, for every compact oriented surfaceΣ of genus g ≥ N , there is quasi-Fuchsian groupΓ ∼= π1(Σ) whose limit set Λ(Γ) ⊂ C ⊂ CP1 iswithin Hausdorff distance ε of γ.
If γ is invariant under ζ 7→ −ζ, and if g is even,we can also arrange for Λ(Γ) to also be invariantunder reflection through the origin.
280
Lemma.For any piecewise smooth Jordan curveγ ⊂ C and any ε > 0, there is a positive integerN such that, for every compact oriented surfaceΣ of genus g ≥ N , there is quasi-Fuchsian groupΓ ∼= π1(Σ) whose limit set Λ(Γ) ⊂ C ⊂ CP1 iswithin Hausdorff distance ε of γ.
If γ is invariant under ζ 7→ −ζ, and if g is even,we can also arrange for Λ(Γ) to also be invariantunder reflection through the origin.
281
Lemma.For any piecewise smooth Jordan curveγ ⊂ C and any ε > 0, there is a positive integerN such that, for every compact oriented surfaceΣ of genus g ≥ N , there is quasi-Fuchsian groupΓ ∼= π1(Σ) whose limit set Λ(Γ) ⊂ C ⊂ CP1 iswithin Hausdorff distance ε of γ.
If γ is invariant under ζ 7→ −ζ, and if g is even,we can also arrange for Λ(Γ) to also be invariantunder reflection through the origin.
282
Lemma.For any piecewise smooth Jordan curveγ ⊂ C and any ε > 0, there is a positive integerN such that, for every compact oriented surfaceΣ of genus g ≥ N , there is quasi-Fuchsian groupΓ ∼= π1(Σ) whose limit set Λ(Γ) ⊂ C ⊂ CP1 iswithin Hausdorff distance ε of γ.
If γ is invariant under ζ 7→ −ζ, and if g is even,we can also arrange for Λ(Γ) to also be invariantunder reflection through the origin.
Ahlfors-Bers: Quasi-conformal mappings
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289
Theorem A.Consider 4-manifolds M = Σ× S2,where Σ compact Riemann surface of genus g .
Then ∀ even g � 0, ∃ family [gt], t ∈ [0, 1], oflocally-conformally-flat classes on M , such that
• ∃ scalar-flat Kahler metric g0 ∈ [g0]; but
• @ almost-Kahler metric g ∈ [g1].
290
Theorem B. Fix an integer k ≥ 2, and thenconsider the 4-manifolds M = (Σ× S2)#kCP2,where Σ compact Riemann surface of genus g .
Then ∀ even g � 0, ∃ family [gt], t ∈ [0, 1], ofanti-self-dual conformal classes on M , such that
• ∃ scalar-flat Kahler metric g0 ∈ [g0]; but
• @ almost-Kahler metric g ∈ [g1].
291