Critical metrics on connected sums of Einstein four-manifolds · Je Viaclovsky Critical metrics on...

Introduction The gluing procedure The building blocks Remarks on the proof

Critical metrics on connected sums ofEinstein four-manifolds

Jeff Viaclovsky

University of Wisconsin

April 4, 2014

Fields Geometric Analysis Colloquium

Jeff Viaclovsky Critical metrics on connected sums of Einstein four-manifolds


Einstein manifolds

Einstein-Hilbert functional in dimension 4:

R(g) = V ol(g)−1/2

∫MRgdVg,

where Rg is the scalar curvature.

Euler-Lagrange equations:

Ric(g) = λ · g,

where λ is a constant.

(M, g) is called an Einstein manifold.



Einstein manifolds

Einstein-Hilbert functional in dimension 4:

R(g) = V ol(g)−1/2

∫MRgdVg,

where Rg is the scalar curvature.

Euler-Lagrange equations:

Ric(g) = λ · g,

where λ is a constant.

(M, g) is called an Einstein manifold.



Orbifold Limits

Theorem (Anderson, Bando-Kasue-Nakajima, Tian)

(Mi, gi) sequence of 4-dimensional Einstein manifolds satisfying∫|Rm|2 < Λ, diam(gi) < D, V ol(gi) > V > 0.

Then for a subsequence {j} ⊂ {i},

(Mj , gj)Cheeger−Gromov−−−−−−−−−−−→ (M∞, g∞),

where (M∞, g∞) is an orbifold with finitely many singular points.



Kummer example

Rescaling such a sequence to have bounded curvature near asingular point yields Ricci-flat non-compact limits calledasympotically locally Euclidean spaces (ALE spaces), also called“bubbles”.

Example

There exists a sequence of Ricci-flat metrics gi on K3 satisfying:

(K3, gi) −→ (T 4/{±1}, gflat).

At each of the 16 singular points, an Eguchi-Hanson metric onT ∗S2 “bubbles off”.



Question

Can you reverse this process?

I.e., start with an Einstein orbifold, “glue on” bubbles at thesingular points, and resolve to a smooth Einstein metric?

In general, answer is “no”.

Reason: this is a self-adjoint gluing problem so possibility ofmoduli is an obstruction.



Question







Question







Question







Self-dual or anti-self-dual metrics

(M4, g) oriented.

R =

W+ + R

12I E

E W− + R12I

.

E = Ric− (R/4)g.

W+ = 0 is called anti-self-dual (ASD).W− = 0 is called self-dual (SD).

Either condition is conformally invariant.




(M4, g) oriented.

R =

W+ + R

12I E

E W− + R12I

.

E = Ric− (R/4)g.






(M4, g) oriented.

R =

W+ + R

12I E

E W− + R12I

.

E = Ric− (R/4)g.





ASD gluing

Theorem (Donaldson-Friedman, Floer, Kovalev-Singer, etc.)

If (M1, g1) and (M2.g2) are ASD and H2(Mi, gi) = {0} then thereexist ASD metrics on the connected sum M1#M2.

Contrast with Einstein gluing problem:

• ASD situation can be unobstructed (H2 = 0), yet still havemoduli (H1 6= 0).

• Cannot happen for a self-adjoint gluing problem.



ASD gluing

Theorem (Donaldson-Friedman, Floer, Kovalev-Singer, etc.)

If (M1, g1) and (M2.g2) are ASD and H2(Mi, gi) = {0} then thereexist ASD metrics on the connected sum M1#M2.

Contrast with Einstein gluing problem:

• ASD situation can be unobstructed (H2 = 0), yet still havemoduli (H1 6= 0).

• Cannot happen for a self-adjoint gluing problem.



Biquard’s Theorem

Recently, Biquard showed the following:

Theorem (Biquard, 2011)

Let (M, g) be a (non-compact) Poincare-Einstein (P-E) metricwith a Z/2Z orbifold singularity at p ∈M . If (M, g) is rigid, thenthe singularity can be resolved to a P-E Einstein metric by gluingon an Eguchi-Hanson metric if and only if

det(R+)(p) = 0.

Self-adjointness of this gluing problem is overcome by freedom ofchoosing the boundary conformal class of the P-E metric.



Biquard’s Theorem

Recently, Biquard showed the following:

Theorem (Biquard, 2011)

Let (M, g) be a (non-compact) Poincare-Einstein (P-E) metricwith a Z/2Z orbifold singularity at p ∈M . If (M, g) is rigid, thenthe singularity can be resolved to a P-E Einstein metric by gluingon an Eguchi-Hanson metric if and only if

det(R+)(p) = 0.

Self-adjointness of this gluing problem is overcome by freedom ofchoosing the boundary conformal class of the P-E metric.



Quadratic curvature functionals

A basis for the space of quadratic curvature functionals is

W =

∫|W |2 dV, ρ =

∫|Ric|2 dV, S =

∫R2 dV.

In dimension four, the Chern-Gauss-Bonnet formula

32π2χ(M) =

∫|W |2 dV − 2

∫|Ric|2 dV +

2

3

∫R2 dV

implies that ρ can be written as a linear combination of the othertwo (plus a topological term).Consequently, we will be interested in the functional

Bt[g] =

∫|W |2 dV + t

∫R2 dV.





W =

∫|W |2 dV, ρ =

∫|Ric|2 dV, S =

∫R2 dV.


32π2χ(M) =

∫|W |2 dV − 2

∫|Ric|2 dV +

2

3

∫R2 dV

implies that ρ can be written as a linear combination of the othertwo (plus a topological term).

Consequently, we will be interested in the functional

Bt[g] =

∫|W |2 dV + t

∫R2 dV.





W =

∫|W |2 dV, ρ =

∫|Ric|2 dV, S =

∫R2 dV.


32π2χ(M) =

∫|W |2 dV − 2

∫|Ric|2 dV +

2

3

∫R2 dV

implies that ρ can be written as a linear combination of the othertwo (plus a topological term).Consequently, we will be interested in the functional

Bt[g] =

∫|W |2 dV + t

∫R2 dV.



Generalization of the Einstein condition

The Euler-Lagrange equations of Bt are given by

Bt ≡ B + tC = 0,

where B is the Bach tensor defined by

Bij ≡ −4(∇k∇lWikjl +

1

2RklWikjl

),

and C is the tensor defined by

Cij = 2∇i∇jR− 2(∆R)gij − 2RRij +1

2R2gij .

• Any Einstein metric is critical for Bt.• We will refer to such a critical metric as a Bt-flat metric.





Bt ≡ B + tC = 0,



1

2RklWikjl

),



2R2gij .

• Any Einstein metric is critical for Bt.

• We will refer to such a critical metric as a Bt-flat metric.





Bt ≡ B + tC = 0,



1

2RklWikjl

),



2R2gij .

• Any Einstein metric is critical for Bt.• We will refer to such a critical metric as a Bt-flat metric.




For t 6= 0, by taking a trace of the E-L equations:

∆R = 0.

If M is compact, this implies R = constant.

Consequently, the Bt-flat condition is equivalent to

B = 2tR · E ,

where E denotes the traceless Ricci tensor.

• The Bach tensor is a constant multiple of the traceless Riccitensor.





∆R = 0.



B = 2tR · E ,







∆R = 0.



B = 2tR · E ,





Orbifold Limits

The Bt-flat equation can be rewritten as

∆Ric = Rm ∗Rc. (∗)

Theorem (Tian-V)

(Mi, gi) sequence of 4-dimensional manifolds satisfying (∗) and∫|Rm|2 < Λ, V ol(B(q, s)) > V s4, b1(Mi) < B.

Then for a subsequence {j} ⊂ {i},

(Mj , gj)Cheeger−Gromov−−−−−−−−−−−→ (M∞, g∞),

where (M∞, g∞) is a multi-fold satisyfing (∗), with finitely manysingular points.



Question


I.e., start with an critical orbifold, “glue on” critical bubbles at thesingular points, and resolve to a smooth critical metric?

Answer is still “no” in general, because this is also a self-adjointgluing problem.

Our main theorem: the answer is “YES” in certain cases.



Question







Question







Question







Main theorem

Theorem (Gursky-V 2013)

A Bt-flat metric exists on the manifolds in the table for some tnear the indicated value of t0.

Table: Simply-connected examples with one bubble

Topology of connected sum Value(s) of t0

CP2#CP2 −1/3

S2 × S2#CP2= CP2#2CP2 −1/3, −(9m1)−1

2#S2 × S2 −2(9m1)−1

The constant m1 is a geometric invariant called the mass of ancertain asymptotically flat metric: the Green’s function metric ofthe product metric S2 × S2.



Remarks

• CP2#CP2admits an U(2)-invariant Einstein metric called the

Page metric. Does not admit any Kahler-Einstein metric, butthe Page metric is conformal to an extremal Kahler metric.

• CP2#2CP2admits a toric invariant Einstein metric called the

Chen-LeBrun-Weber metric. Again, does not admit anyKahler-Einstein metric, but the Chen-LeBrun-Weber metric isconformal to an extremal Kahler metric.

• S2 × S2#S2 × S2 does not admit any Kahler metric, it doesnot even admit an almost complex structure. Our metric is thefirst known example of a “canonical” metric on this manifold.



Remarks








Remarks








Green’s function metric

The conformal Laplacian:

Lu = −6∆u+Ru.

If (M, g) is compact and R > 0, then for any p ∈M , there is aunique positive solution to the equation

LG = 0 on M \ {p}G = ρ−2(1 + o(1))

as ρ→ 0, where ρ is geodesic distance to the basepoint p.

• Denote N = M \ {p} with metric gN = G2gM . The metricgN is scalar-flat and asymptotically flat of order 2.

• If (M, g) is Bach-flat, then (N, gN ) is also Bach-flat (fromconformal invariance) and scalar-flat (since we used theGreen’s function). Consequently, gN is Bt-flat for all t ∈ R.





Lu = −6∆u+Ru.


LG = 0 on M \ {p}G = ρ−2(1 + o(1))








Lu = −6∆u+Ru.


LG = 0 on M \ {p}G = ρ−2(1 + o(1))






The approximate metric

• Let (Z, gZ) and (Y, gY ) be Einstein manifolds, and assumethat gY has positive scalar curvature.

• Choose basepoints z0 ∈ Z and y0 ∈ Y .

• Convert (Y, gY ) into an asymptotically flat (AF) metric(N, gN ) using the Green’s function for the conformalLaplacian based at y0. As pointed out above, gN is Bt-flat forany t.

• Let a > 0 be small, and consider Z \B(z0, a). Scale thecompact metric to (Z, g = a−4gZ). Attach this metric to themetric (N \B(a−1), gN ) using cutoff functions near theboundary, to obtain a smooth metric on the connect sumZ#Y .

























Damage zone

AF metric

Compact Einsteinmetric

Figure: The approximate metric.

Since both gZ and gN are Bt-flat, this metric is an “approximate”Bt-flat metric, with vanishing Bt tensor away from the “damagezone”, where cutoff functions were used.



Gluing parameters

In general, there are several degrees of freedom in this approximatemetric.

• The scaling parameter a (1-dimensional).

• Rotational freedom when attaching (6-dimensional).

• Freedom to move the base points of either factor(8-dimensional).

Total of 15 gluing parameters.



Gluing parameters


• The scaling parameter a

(1-dimensional).






Gluing parameters








Gluing parameters



• Rotational freedom when attaching

(6-dimensional).





Gluing parameters








Gluing parameters




• Freedom to move the base points of either factor

(8-dimensional).




Gluing parameters








Gluing parameters








Lyapunov-Schmidt reduction

These 15 gluing parameters yield a 15-dimensional space of“approximate” kernel of the linearized operator. Using aLyapunov-Schmidt reduction argument, one can reduce theproblem to that of finding a zero of the Kuranishi map

Ψ : U ⊂ R15 → R15.

• It is crucial to use certain weighted norms to find a boundedright inverse for the linearized operator.

• This 15-dimensional problem is too difficult in general: we willtake advantage of various symmetries in order to reduce toonly 1 free parameter: the scaling parameter a.





Ψ : U ⊂ R15 → R15.







Ψ : U ⊂ R15 → R15.





Technical theorem

The leading term of the Kuranishi map corresponding to thescaling parameter is given by:

Theorem (Gursky-V 2013)

As a→ 0, then for any ε > 0,

Ψ1 =(2

3W (z0) ~W (y0) + 4tR(z0)mass(gN )

)ω3a

4 +O(a6−ε),

where ω3 = V ol(S3), and the product of the Weyl tensors is givenby

W (z0) ~W (y0) =∑ijkl

Wijkl(z0)(Wijkl(y0) +Wilkj(y0)),

where Wijkl(·) denotes the components of the Weyl tensor in anormal coordinate system at the corresponding point.



The Fubini-Study metric

(CP2, gFS), the Fubini-Study metric, Ric = 6g.

Torus action:

[z0, z1, z2] 7→ [z0, eiθ1z1, e

iθ2z2].

Flip symmetry:

[z0, z1, z2] 7→ [z0, z2, z1].





Torus action:

[z0, z1, z2] 7→ [z0, eiθ1z1, e

iθ2z2].

Flip symmetry:

[z0, z1, z2] 7→ [z0, z2, z1].





Torus action:

[z0, z1, z2] 7→ [z0, eiθ1z1, e

iθ2z2].

Flip symmetry:

[z0, z1, z2] 7→ [z0, z2, z1].




[1,0,0] [0,1,0]

[0,0,1]

Figure: Orbit space of the torus action on CP2.



The product metric

(S2 × S2, gS2×S2), the product of 2-dimensional spheres ofGaussian curvature 1, Ric = g.

Torus action:

Product of rotations fixing north and south poles.

Flip symmetry:

(p1, p2) 7→ (p2, p1).



The product metric


Torus action:


Flip symmetry:

(p1, p2) 7→ (p2, p1).



The product metric


Torus action:


Flip symmetry:

(p1, p2) 7→ (p2, p1).



The product metric

(n,n) (n,s)

(s,s)(s,n)

Figure: Orbit space of the torus action on S2 × S2.



Mass of Green’s function metric

Recall the mass of an AF space is defined by

mass(gN ) = limR→∞

ω−13

∫S(R)

∑i,j

(∂igij − ∂jgii)(∂i y dV ),

with ω3 = V ol(S3).

The Green’s function metric of the Fubini-Study metric gFS is alsoknown as the Burns metric, and is completely explicit, with massgiven by

mass(gFS) = 2.




Recall the mass of an AF space is defined by

mass(gN ) = limR→∞

ω−13

∫S(R)

∑i,j

(∂igij − ∂jgii)(∂i y dV ),

with ω3 = V ol(S3).

The Green’s function metric of the Fubini-Study metric gFS is alsoknown as the Burns metric, and is completely explicit, with massgiven by

mass(gFS) = 2.




However, the Green’s function metric gS2×S2 of the product metricdoes not seem to have a known explicit description. We will denote

m1 = mass(gS2×S2).

By the positive mass theorem of Schoen-Yau, m1 > 0. Note thatsince S2 × S2 is spin, this also follows from Witten’s proof of thepositive mass theorem.

Remark

For the curious, the mass m1 = .5872...., which implies that(−9m1)−1 = −.1892.....




However, the Green’s function metric gS2×S2 of the product metricdoes not seem to have a known explicit description. We will denote

m1 = mass(gS2×S2).

By the positive mass theorem of Schoen-Yau, m1 > 0. Note thatsince S2 × S2 is spin, this also follows from Witten’s proof of thepositive mass theorem.

Remark

For the curious, the mass m1 = .5872...., which implies that(−9m1)−1 = −.1892.....



Remarks on the proof

• We impose the toric symmetry and “flip” symmetry in orderto reduce the number of free parameters to 1 (only the scalingparameter). That is, we perform an equivariant gluing.

• The special value of t0 is computed by

2

3W (z0) ~W (y0) + 4t0R(z0)mass(gN ) = 0.

• This choice of t0 makes the leading term of the Kuranishimap vanish, and is furthermore a nondegenerate zero (ifR(z0) > 0; mass(gN ) > 0 by the positive mass theorem).






2

3W (z0) ~W (y0) + 4t0R(z0)mass(gN ) = 0.







2

3W (z0) ~W (y0) + 4t0R(z0)mass(gN ) = 0.




First case



CP2#CP2 −1/3

• The compact metric is the Fubini-Study metric, with a BurnsAF metric glued on, a computation yields t0 = −1/3.



Second case



S2 × S2#CP2= CP2#2CP2 −1/3, −(9m1)−1

• The compact metric is the product metric on S2 × S2, with aBurns AF metric glued on, this gives t0 = −1/3.

• Alternatively, take the compact metric to be (CP2, gFS), witha Green’s function S2 × S2 metric glued on. In this case,t0 = −(9m1)−1.



Second case



S2 × S2#CP2= CP2#2CP2 −1/3, −(9m1)−1

• The compact metric is the product metric on S2 × S2, with aBurns AF metric glued on, this gives t0 = −1/3.

• Alternatively, take the compact metric to be (CP2, gFS), witha Green’s function S2 × S2 metric glued on. In this case,t0 = −(9m1)−1.



Third case


Topology of connected sum Value(s) of t02#S2 × S2 −2(9m1)−1

• The compact metric is the product metric on S2 × S2, with aGreen’s function S2 × S2 metric glued on. In this case,t0 = −2(9m1)−1.



Other symmetries

By imposing other symmetries, we can perform the gluingoperation with more than one bubble:

Table: Simply-connected examples with several bubbles

Topology of connected sum Value of t0 Symmetry

3#S2 × S2 −2(9m1)−1 bilateral

S2 × S2#2CP2= CP2#3CP2 −1/3 bilateral

CP2#3CP2 −1/3 trilateral

CP2#3(S2 × S2) = 4CP2#3CP2 −(9m1)−1 trilateral

S2 × S2#4CP2= CP2#5CP2 −1/3 quadrilateral

5#S2 × S2 −2(9m1)−1 quadrilateral



Non-simply-connected examples

We can also use the quotient spaces G(2, 4) and RP2 × RP2 asbuilding blocks to obtain non-simply connected examples. We donot list all of the examples here, but just note that we find acritical metric on G(2, 4)#G(2, 4), which has infinite fundamentalgroup, and therefore does not admit any positive Einstein metricby Myers’ Theorem.



Technical Points

• Ellipticity and gauging. The Bt-flat equations are not ellipticdue to diffeomorphism invariance. A gauging procedureanalogous to the Bianchi gauge is used.

• Rigidity of gFS and gS2×S2 . Proved recently by Gursky-V (toappear in Crelle’s Journal). Extends earlier work of O.Kobayashi for the Bach tensor, and N. Koiso for the Einsteinequations.

• Refined approximate metric. The approximate metricdescribed above is not good enough. Can be improved bymatching up leading terms of the metrics by solving certainauxiliary linear equations, so that the cutoff functiondisappears from the leading term. This step is inspired by thework of O. Biquard mentioned above.



Technical Points






Technical Points






Ellipticity and gauging

The linearized operator of the Bt-flat equation is not elliptic, dueto diffeomorphism invariance. However, consider the “gauged”nonlinear map P given by

Pg(θ) = (B + tC)(g + θ) +Kg+θ[δgKgδg◦θ],

where Kg denotes the conformal Killing operator,

(Kgω)ij = ∇iωj +∇jωi −1

2(δgω)gij ,

δ denotes the divergence operator,

(δgh)j = ∇ihij ,

and◦θ = θ − 1

4trgθg,

is the traceless part of θ.Jeff Viaclovsky Critical metrics on connected sums of Einstein four-manifolds



Let St ≡ P ′(0) denote the linearized operator at θ = 0.

Proposition

If t 6= 0, then St is elliptic. Furthermore, if P (θ) = 0, andθ ∈ C4,α for some 0 < α < 1, then Bt(g + θ) = 0 and θ ∈ C∞.

• Proof is an integration-by-parts. Uses crucially that theBt-flat equations are variational (recall Bt is the functional),so δBt = 0. Equivalent to diffeomorphism invariance of Bt.




Let St ≡ P ′(0) denote the linearized operator at θ = 0.

Proposition

If t 6= 0, then St is elliptic. Furthermore, if P (θ) = 0, andθ ∈ C4,α for some 0 < α < 1, then Bt(g + θ) = 0 and θ ∈ C∞.

• Proof is an integration-by-parts. Uses crucially that theBt-flat equations are variational (recall Bt is the functional),so δBt = 0. Equivalent to diffeomorphism invariance of Bt.



Rigidity

For h transverse-traceless (TT), the linearized operator at anEinstein metric is given by

Sth =(

∆L +1

2R)(

∆L +(1

3+ t)R)h,

where ∆L is the Lichnerowicz Laplacian, defined by

∆Lhij = ∆hij + 2Ripjqhpq − 1

2Rhij .

• This formula was previously obtained for the linearized Bachtensor (t = 0) by O. Kobayashi.

• N. Koiso previously studied infinitesimal Einstein deformationsgiven by TT kernel of the operator ∆L + 1

2R



Rigidity


Sth =(

∆L +1

2R)(

∆L +(1

3+ t)R)h,



2Rhij .



2R



Rigidity


Sth =(

∆L +1

2R)(

∆L +(1

3+ t)R)h,



2Rhij .



2R



Rigidity

For h = fg, we have

trg(Sth) = 6t(3∆ +R)(∆f). (1)

The rigidity question is then reduced to a separate analysis of theeigenvalues of ∆L on transverse-traceless tensors, and of ∆ onfunctions.

Theorem (Gursky-V)

On (CP2, gFS), H1t = {0} provided that t < 1.

Theorem (Gursky-V)

On (S2 × S2, gS2×S2), H1t = {0} provided that t < 2/3 and

t 6= −1/3. If t = −1/3, then H1t is one-dimensional and spanned

by the element g1 − g2.



Rigidity

For h = fg, we have

trg(Sth) = 6t(3∆ +R)(∆f). (1)


Theorem (Gursky-V)


Theorem (Gursky-V)






Rigidity

For h = fg, we have

trg(Sth) = 6t(3∆ +R)(∆f). (1)


Theorem (Gursky-V)


Theorem (Gursky-V)






Rigidity

• Positive mass theorem says that t0 < 0, so luckily we are inthe rigidity range of the factors.

• Gauge term is carefully chosen so that solutions of thelinearized equation must be in the transverse-traceless gauge.That is, if Sth = 0 then

(Bt)′(h) +KδKδ◦h = 0

implies that separately,

(Bt)′(h) = 0 and δ◦h = 0.



Rigidity

• Positive mass theorem says that t0 < 0, so luckily we are inthe rigidity range of the factors.

• Gauge term is carefully chosen so that solutions of thelinearized equation must be in the transverse-traceless gauge.That is, if Sth = 0 then

(Bt)′(h) +KδKδ◦h = 0

implies that separately,

(Bt)′(h) = 0 and δ◦h = 0.



Refined approximate metric

Let (Z, gZ) be the compact metric. In Riemannian normalcoordinates,

(gZ)ij(z) = δij −1

3Rikjl(z0)zkzl +O(|z|4)ij

as z → z0.

Let (N, gN ) be the Green’s function metric of (Y, gY ), then wehave

(gN )ij(x) = δij −1

3Rikjl(y0)

xkxl

|x|4+ 2A

1

|x|2δij +O(|x|−4+ε)

as |x| → ∞, for any ε > 0.

• The constant A is given by mass(gN ) = 12A−R(y0)/12.







as z → z0.Let (N, gN ) be the Green’s function metric of (Y, gY ), then wehave


3Rikjl(y0)

xkxl

|x|4+ 2A

1

|x|2δij +O(|x|−4+ε)

as |x| → ∞, for any ε > 0.








as z → z0.Let (N, gN ) be the Green’s function metric of (Y, gY ), then wehave


3Rikjl(y0)

xkxl

|x|4+ 2A

1

|x|2δij +O(|x|−4+ε)

as |x| → ∞, for any ε > 0.






3Rikjl(y0)

xkxl

|x|4+ 2A

1

|x|2δij +O(|x|−4+ε).

We consider a−4gZ and let z = a2x, then we have

a−4(gZ)ij(x) = δij − a4 1

3Rikjl(z0)xkxl + · · · .

• Second order terms do not agree. Need to construct newmetrics on the factors so that these terms agree. This is doneby solving the linearized equation on each factor withprescribed leading term the second order term of the othermetric.

• Linear equation on AF metric is obstructed, and this is howthe leading term of the Kuranishi map is computed.





3Rikjl(y0)

xkxl

|x|4+ 2A

1

|x|2δij +O(|x|−4+ε).










3Rikjl(y0)

xkxl

|x|4+ 2A

1

|x|2δij +O(|x|−4+ε).








The obstruction

On (N, gN ), solve

Sh = 0

h = −a4 1

3Rikjl(z0)xkxl +O(|x|ε),

as x→∞.

This equation is obstructed! Instead solve modified equation

Sh = λ · k1,

where k1 pairs nontrivially with the decaying cokernel on the AFspace o1.

Similar procedure on compact piece, except this is unobstructed(compact piece is rigid).



The obstruction

On (N, gN ), solve

Sh = 0

h = −a4 1


as x→∞.


Sh = λ · k1,





The obstruction

On (N, gN ), solve

Sh = 0

h = −a4 1


as x→∞.


Sh = λ · k1,





Computation of leading term

λ = limr→∞

∫B(r)〈Sh, o1〉

= limr→∞

( spherical boundary integrals)

This limit can be computed explicitly using the expansion

(o1)ij =2

3Wikjl(y0)

xkxl

|x|4+ fgij +O(|x|−4+ε)

as x→∞, where f satisfies

∆f = −1

3〈Ric, o1〉,

and recalling that

h = −a4 1


as x→∞.



Computation of leading term

λ = limr→∞

∫B(r)〈Sh, o1〉

= limr→∞

( spherical boundary integrals)

This limit can be computed explicitly using the expansion

(o1)ij =2

3Wikjl(y0)

xkxl

|x|4+ fgij +O(|x|−4+ε)

as x→∞, where f satisfies

∆f = −1

3〈Ric, o1〉,

and recalling that

h = −a4 1


as x→∞.Jeff Viaclovsky Critical metrics on connected sums of Einstein four-manifolds


Final remarks

The proof shows that there is a dichotomy.

Either

• (i) there is a critical metric at exactly the critical t0, in whichcase there would necessarily be a 1-dimensional moduli spaceof solutions for this fixed t0, or

• (ii) for each value of the gluing parameter a sufficiently small,there will be a critical metric for a corresponding value oft0 = t0(a).

To determine which case actually happens, one must calculate thenext term in the expansion of the Kuranishi map.



Final remarks

The proof shows that there is a dichotomy. Either

• (i) there is a critical metric at exactly the critical t0, in whichcase there would necessarily be a 1-dimensional moduli spaceof solutions for this fixed t0,

or





Final remarks







Final remarks






Bonus slide: the mass of S2 × S2

m1 =2

3− 12

∞∑j,k=0

(2j + 1)(2k + 1)

λj,k

( j∑p=0

k∑q=0

cp,qj,k fp,q

),

where

λj,k = j(j + 1) + k(k + 1) +2

3,

cp,qj,k = (−1)p+q(j

p

)(j + p

p

)(k

q

)(k + q

q

),

fp,q = − 1

108

1

p+ q + 3

{29 + 32(p+ q) + 21(p2 + q2) + 6(p3 + q3)

+ 4(−1)p+1(p+ 1)(p+ 2)(3p2 + 9p+ 10)(

log(2)−A(p+ 2))

+ 4(−1)q+1(q + 1)(q + 2)(3q2 + 9q + 10)(

log(2)−A(q + 2)),

where

A(p) =

p∑i=1

(−1)i−1 1

i.

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