Introduction The gluing procedure The building blocks Remarks on the proof Critical metrics on connected sums of Einstein four-manifolds Jeff Viaclovsky University of Wisconsin, Madison March 22, 2013 Kyoto Jeff Viaclovsky Critical metrics on connected sums of Einstein four-manifolds
Introduction The gluing procedure The building blocks Remarks on the proof Einstein manifolds Einstein-Hilbert functional in dimension 4 : � ˜ R ( g ) = V ol ( g ) − 1 / 2 R g dV g , M where R g is the scalar curvature. Jeff Viaclovsky Critical metrics on connected sums of Einstein four-manifolds
Introduction The gluing procedure The building blocks Remarks on the proof Einstein manifolds Einstein-Hilbert functional in dimension 4 : � ˜ R ( g ) = V ol ( g ) − 1 / 2 R g dV g , M where R g is the scalar curvature. Euler-Lagrange equations: Ric ( g ) = λ · g, where λ is a constant. ( M, g ) is called an Einstein manifold . Jeff Viaclovsky Critical metrics on connected sums of Einstein four-manifolds
Introduction The gluing procedure The building blocks Remarks on the proof Orbifold Limits Theorem (Anderson, Bando-Kasue-Nakajima, Tian) ( M i , g i ) sequence of 4 -dimensional Einstein manifolds satisfying � | Rm | 2 < Λ , diam( g i ) < D, V ol ( g i ) > V > 0 . Then for a subsequence { j } ⊂ { i } , Cheeger − Gromov ( M j , g j ) − − − − − − − − − − − → ( M ∞ , g ∞ ) , where ( M ∞ , g ∞ ) is an orbifold with finitely many singular points. Jeff Viaclovsky Critical metrics on connected sums of Einstein four-manifolds
Introduction The gluing procedure The building blocks Remarks on the proof Kummer example Rescaling such a sequence to have bounded curvature near a singular point yields Ricci-flat non-compact limits called asympotically locally Euclidean spaces (ALE spaces), also called “bubbles”. Example There exists a sequence of Ricci-flat metrics g i on K 3 satisyfing: → ( T 4 / {± 1 } , g flat ) . ( K 3 , g i ) − At each of the 16 singular points, an Eguchi-Hanson metric on T ∗ S 2 “bubbles off”. Jeff Viaclovsky Critical metrics on connected sums of Einstein four-manifolds
Introduction The gluing procedure The building blocks Remarks on the proof Question Can you reverse this process? Jeff Viaclovsky Critical metrics on connected sums of Einstein four-manifolds
Introduction The gluing procedure The building blocks Remarks on the proof Question Can you reverse this process? I.e., start with an Einstein orbifold, “glue on” bubbles at the singular points, and resolve to a smooth Einstein metric? Jeff Viaclovsky Critical metrics on connected sums of Einstein four-manifolds
Introduction The gluing procedure The building blocks Remarks on the proof Question Can you reverse this process? I.e., start with an Einstein orbifold, “glue on” bubbles at the singular points, and resolve to a smooth Einstein metric? In general, answer is “no”. Jeff Viaclovsky Critical metrics on connected sums of Einstein four-manifolds
Introduction The gluing procedure The building blocks Remarks on the proof Question Can you reverse this process? I.e., start with an Einstein orbifold, “glue on” bubbles at the singular points, and resolve to a smooth Einstein metric? In general, answer is “no”. Reason: this is a self-adjoint gluing problem so possibility of moduli is an obstruction. Jeff Viaclovsky Critical metrics on connected sums of Einstein four-manifolds
Introduction The gluing procedure The building blocks Remarks on the proof Self-dual or anti-self-dual metrics ( M 4 , g ) oriented. W + + R 12 I E R = . W − + R E 12 I E = Ric − ( R/ 4) g . Jeff Viaclovsky Critical metrics on connected sums of Einstein four-manifolds
Introduction The gluing procedure The building blocks Remarks on the proof Self-dual or anti-self-dual metrics ( M 4 , g ) oriented. W + + R 12 I E R = . W − + R E 12 I E = Ric − ( R/ 4) g . W + = 0 is called anti-self-dual (ASD). W − = 0 is called self-dual (SD). Jeff Viaclovsky Critical metrics on connected sums of Einstein four-manifolds
Introduction The gluing procedure The building blocks Remarks on the proof Self-dual or anti-self-dual metrics ( M 4 , g ) oriented. W + + R 12 I E R = . W − + R E 12 I 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. Jeff Viaclovsky Critical metrics on connected sums of Einstein four-manifolds
Introduction The gluing procedure The building blocks Remarks on the proof ASD gluing Theorem (Donaldson-Friedman, Floer, Kovalev-Singer, etc.) If ( M 1 , g 1 ) and ( M 2 .g 2 ) are ASD and H 2 ( M i , g i ) = { 0 } then there exists ASD metrics on the connected sum M 1 # M 2 . Jeff Viaclovsky Critical metrics on connected sums of Einstein four-manifolds
Introduction The gluing procedure The building blocks Remarks on the proof ASD gluing Theorem (Donaldson-Friedman, Floer, Kovalev-Singer, etc.) If ( M 1 , g 1 ) and ( M 2 .g 2 ) are ASD and H 2 ( M i , g i ) = { 0 } then there exists ASD metrics on the connected sum M 1 # M 2 . Contrast with Einstein gluing problem: • ASD situation can be unobstructed ( H 2 = 0 ), yet still have moduli ( H 1 � = 0) . • Cannot happen for a self-adjoint gluing problem. Jeff Viaclovsky Critical metrics on connected sums of Einstein four-manifolds
Introduction The gluing procedure The building blocks Remarks on the proof Biquard’s Theorem Recently, Biquard showed the following: Theorem (Biquard, 2011) Let ( M, g ) be a (non-compact) Poincar´ e-Einstein (P-E) metric with a Z / 2 Z orbifold singularity at p ∈ M . If ( M, g ) is rigid, then the singularity can be resolved to a P-E Einstein metric by gluing on an Eguchi-Hanson metric if and only if det( R + )( p ) = 0 . Jeff Viaclovsky Critical metrics on connected sums of Einstein four-manifolds
Introduction The gluing procedure The building blocks Remarks on the proof Biquard’s Theorem Recently, Biquard showed the following: Theorem (Biquard, 2011) Let ( M, g ) be a (non-compact) Poincar´ e-Einstein (P-E) metric with a Z / 2 Z orbifold singularity at p ∈ M . If ( M, g ) is rigid, then the singularity can be resolved to a P-E Einstein metric by gluing on an Eguchi-Hanson metric if and only if det( R + )( p ) = 0 . Self-adjointness of this gluing problem is overcome by freedom of choosing the boundary conformal class of the P-E metric. Jeff Viaclovsky Critical metrics on connected sums of Einstein four-manifolds
Introduction The gluing procedure The building blocks Remarks on the proof Quadratic curvature functionals A basis for the space of quadratic curvature functionals is � � � | W | 2 dV, | Ric | 2 dV, R 2 dV. W = ρ = S = Jeff Viaclovsky Critical metrics on connected sums of Einstein four-manifolds
Introduction The gluing procedure The building blocks Remarks on the proof Quadratic curvature functionals A basis for the space of quadratic curvature functionals is � � � | W | 2 dV, | Ric | 2 dV, R 2 dV. W = ρ = S = In dimension four, the Chern-Gauss-Bonnet formula � � | Ric | 2 dV + 2 � | W | 2 dV − 2 R 2 dV 32 π 2 χ ( M ) = 3 implies that ρ can be written as a linear combination of the other two (plus a topological term). Jeff Viaclovsky Critical metrics on connected sums of Einstein four-manifolds
Introduction The gluing procedure The building blocks Remarks on the proof Quadratic curvature functionals A basis for the space of quadratic curvature functionals is � � � | W | 2 dV, | Ric | 2 dV, R 2 dV. W = ρ = S = In dimension four, the Chern-Gauss-Bonnet formula � � | Ric | 2 dV + 2 � | W | 2 dV − 2 R 2 dV 32 π 2 χ ( M ) = 3 implies that ρ can be written as a linear combination of the other two (plus a topological term). Consequently, we will be interested in the functional � � | W | 2 dV + t R 2 dV. B t [ g ] = Jeff Viaclovsky Critical metrics on connected sums of Einstein four-manifolds
Introduction The gluing procedure The building blocks Remarks on the proof Generalization of the Einstein condition The Euler-Lagrange equations of B t are given by B t ≡ B + tC = 0 , where B is the Bach tensor defined by ∇ k ∇ l W ikjl + 1 � � 2 R kl W ikjl B ij ≡ − 4 = 0 , and C is the tensor defined by C ij = 2 ∇ i ∇ j R − 2(∆ R ) g ij − 2 RR ij + 1 2 R 2 g ij . Jeff Viaclovsky Critical metrics on connected sums of Einstein four-manifolds
Introduction The gluing procedure The building blocks Remarks on the proof Generalization of the Einstein condition The Euler-Lagrange equations of B t are given by B t ≡ B + tC = 0 , where B is the Bach tensor defined by ∇ k ∇ l W ikjl + 1 � � 2 R kl W ikjl B ij ≡ − 4 = 0 , and C is the tensor defined by C ij = 2 ∇ i ∇ j R − 2(∆ R ) g ij − 2 RR ij + 1 2 R 2 g ij . • Any Einstein metric is critical for B t . Jeff Viaclovsky Critical metrics on connected sums of Einstein four-manifolds
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