Some Progress in Conformal Geometry

This is a survey paper of our current research on the theory of partial differential equations in conformal geometry. Our intention is to describe some of our current works in a rather brief and expository fashion. We are not giving a comprehensive survey on the subject and references cited here are...

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Published in:	Symmetry, Integrability and Geometry: Methods and Applications
Date:	2007
Main Authors:	Sun-Yung A. Chang, Qing, J., Yang, P.
Format:	Article
Language:	English
Published:	Інститут математики НАН України 2007
Online Access:	https://nasplib.isofts.kiev.ua/handle/123456789/147215
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Cite this:	Some Progress in Conformal Geometry / Sun-Yung A. Chang, J. Qing, P. Yang // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 15 назв. — англ.

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spelling	Sun-Yung A. Chang Qing, J. Yang, P. 2019-02-13T19:11:07Z 2019-02-13T19:11:07Z 2007 Some Progress in Conformal Geometry / Sun-Yung A. Chang, J. Qing, P. Yang // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 15 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 53A30; 53C20; 35J60 https://nasplib.isofts.kiev.ua/handle/123456789/147215 This is a survey paper of our current research on the theory of partial differential equations in conformal geometry. Our intention is to describe some of our current works in a rather brief and expository fashion. We are not giving a comprehensive survey on the subject and references cited here are not intended to be complete. We introduce a bubble tree structure to study the degeneration of a class of Yamabe metrics on Bach flat manifolds satisfying some global conformal bounds on compact manifolds of dimension 4. As applications, we establish a gap theorem, a finiteness theorem for diffeomorphism type for this class, and diameter bound of the σ2-metrics in a class of conformal 4-manifolds. For conformally compact Einstein metrics we introduce an eigenfunction compactification. As a consequence we obtain some topological constraints in terms of renormalized volumes. This paper is a contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Some Progress in Conformal Geometry Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	Some Progress in Conformal Geometry
spellingShingle	Some Progress in Conformal Geometry Sun-Yung A. Chang Qing, J. Yang, P.
title_short	Some Progress in Conformal Geometry
title_full	Some Progress in Conformal Geometry
title_fullStr	Some Progress in Conformal Geometry
title_full_unstemmed	Some Progress in Conformal Geometry
title_sort	some progress in conformal geometry
author	Sun-Yung A. Chang Qing, J. Yang, P.
author_facet	Sun-Yung A. Chang Qing, J. Yang, P.
publishDate	2007
language	English
container_title	Symmetry, Integrability and Geometry: Methods and Applications
publisher	Інститут математики НАН України
format	Article
description	This is a survey paper of our current research on the theory of partial differential equations in conformal geometry. Our intention is to describe some of our current works in a rather brief and expository fashion. We are not giving a comprehensive survey on the subject and references cited here are not intended to be complete. We introduce a bubble tree structure to study the degeneration of a class of Yamabe metrics on Bach flat manifolds satisfying some global conformal bounds on compact manifolds of dimension 4. As applications, we establish a gap theorem, a finiteness theorem for diffeomorphism type for this class, and diameter bound of the σ2-metrics in a class of conformal 4-manifolds. For conformally compact Einstein metrics we introduce an eigenfunction compactification. As a consequence we obtain some topological constraints in terms of renormalized volumes.
issn	1815-0659
url	https://nasplib.isofts.kiev.ua/handle/123456789/147215
citation_txt	Some Progress in Conformal Geometry / Sun-Yung A. Chang, J. Qing, P. Yang // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 15 назв. — англ.
work_keys_str_mv	AT sunyungachang someprogressinconformalgeometry AT qingj someprogressinconformalgeometry AT yangp someprogressinconformalgeometry
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 3 (2007), 122, 17 pages Some Progress in Conformal Geometry? Sun-Yung A. CHANG †, Jie QING ‡ and Paul YANG † † Department of Mathematics, Princeton University, Princeton, NJ 08540, USA E-mail: chang@math.princeton.edu, yang@math.princeton.edu ‡ Department of Mathematics, University of California, Santa Cruz, Santa Cruz, CA 95064, USA E-mail: qing@ucsc.edu Received August 30, 2007, in final form December 07, 2007; Published online December 17, 2007 Original article is available at http://www.emis.de/journals/SIGMA/2007/122/ Abstract. This is a survey paper of our current research on the theory of partial differen- tial equations in conformal geometry. Our intention is to describe some of our current works in a rather brief and expository fashion. We are not giving a comprehensive survey on the subject and references cited here are not intended to be complete. We introduce a bubble tree structure to study the degeneration of a class of Yamabe metrics on Bach flat manifolds satisfying some global conformal bounds on compact manifolds of dimension 4. As applications, we establish a gap theorem, a finiteness theorem for diffeomorphism type for this class, and diameter bound of the σ2-metrics in a class of conformal 4-manifolds. For conformally compact Einstein metrics we introduce an eigenfunction compactification. As a consequence we obtain some topological constraints in terms of renormalized volumes. Key words: Bach flat metrics; bubble tree structure; degeneration of metrics; conformally compact; Einstein; renormalized volume 2000 Mathematics Subject Classification: 53A30; 53C20; 35J60 Dedicated to the memory of Thomas Branson 1 Conformal gap and finiteness theorem for a class of closed 4-manifolds 1.1 Introduction Suppose that (M4, g) is a closed 4-manifold. It follows from the positive mass theorem that, for a 4-manifold with positive Yamabe constant,∫ M σ2dv ≤ 16π2 and equality holds if and only if (M4, g) is conformally equivalent to the standard 4-sphere, where σ2[g] = 1 24 R2 − 1 2 \|E\|2, R is the scalar curvature of g and E is the traceless Ricci curvature of g. This is an interesting fact in conformal geometry because the above integral is a conformal invariant like the Yamabe constant. ?This paper is a contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson. The full collection is available at http://www.emis.de/journals/SIGMA/MGC2007.html mailto:chang@math.princeton.edu mailto:yang@math.princeton.edu mailto:qing@ucsc.edu http://www.emis.de/journals/SIGMA/2007/122/ http://www.emis.de/journals/SIGMA/MGC2007.html 2 S.-Y.A. Chang, J. Qing and P. Yang One may ask, whether there is a constant ε0 > 0 such that a closed 4-manifold M4 has to be diffeomorphic to S4 if it admits a metric g with positive Yamabe constant and∫ M σ2[g]dvg ≥ (1− ε)16π2. for some ε < ε0? Notice that here the Yamabe invariant for such [g] is automatically close to that for the round 4-sphere. There is an analogous gap theorem of Bray and Neves for Yamabe invariant in dimension 3 [4]. One cannot expect the Yamabe invariant alone to isolate the sphere, and it is more plausible to consider the integral of σ2. We will answer the question affirmatively in the class of Bach flat 4-manifolds. Recall that Riemann curvature tensor decomposes into Rijkl = Wijkl + (Aikgjl −Ailgjk −Ajkgil +Ajlgik), in dimension 4, where Wijkl is the Weyl curvature, Aij = 1 2 ( Rij − 1 6 Rgij ) is Weyl–Schouten curvature tensor and Rij is the Ricci curvature tensor. Also recall that the Bach tensor is Bij = Wkijl,lk + 1 2 RklWkijl. We say that a metric g is Bach flat if Bij = 0. Bach flat metrics are critical metrics for the functional ∫ M \|W \|2dv. Bach flatness is conformally invariant in dimension 4. It follows from Chern–Gauss–Bonnet, 8π2χ(M4) = ∫ M (σ2 + \|W \|2)dv, that ∫ M σ2dv is conformally invariant. The gap theorem is as follows: Theorem 1. Suppose that (M4, [g]) is a Bach flat closed 4-manifold with positive Yamabe con- stant and that∫ M (\|W \|2dv)[g] ≤ Λ0 for some fixed positive number Λ0. Then there is a positive number ε0 > 0 such that, if∫ M σ2[g]dvg ≥ (1− ε)16π2 holds for some constant ε < ε0, then (M4, [g]) is conformally equivalent to the standard 4-sphere. Our approach is based on the recent work on the compactness of Bach flat metrics on 4- manifolds of Tian and Viaclovsky [14, 15], and of Anderson [1]. Indeed our work relies on a more precise understanding of the bubbling process near points of curvature concentration. For that purpose we develop the bubble tree structure in a sequence of metrics that describes precisely the concentration of curvature. Our method to develop bubble tree structure is inspired by the work of Anderson and Cheeger [2] on the bubble tree configurations of the degenerations of metrics of bounded Ricci curvature. Our construction is modeled after this work but differs Some Progress in Conformal Geometry 3 in the way that our bubble tree is built from the bubbles at points with the smallest scale of concentration to bubbles with larger scale; while the bubble tree in [2] is constructed from bubbles of large scale to bubbles with smaller scales. The inductive method of construction of our bubble tree is modeled on earlier work of [3, 11, 13] on the study of concentrations of energies in harmonic maps and the scalar curvature equations. As a consequence of the bubble tree construction we are able to obtain the following finite diffeomorphism theorem: Theorem 2. Suppose that A is a collection of Bach flat Riemannian manifolds (M4, g) with positive Yamabe constant, satisfying∫ M (\|W \|2dv)[g] ≤ Λ0, for some fixed positive number Λ0, and∫ M (σ2dv)[g] ≥ σ0, for some fixed positive number σ0. Then there are only finite many diffeomorphism types in A. It is known that in each conformal class of metrics belonging to the family A, there is a metric ḡ = e2wg such that σ2(Aḡ) = 1, which we shall call the σ2 metric. The bubble tree structure in the degeneration of Yamabe metrics in A is also helpful to understand the behavior of the σ2-metrics in A. For example: Theorem 3. For the conformal classes [g0] ∈ A the conformal metrics g = e2wg0 satisfying the equation σ2(g) = 1 has a uniform bound for the diameter. The detailed version of this work has appeared in our paper [5]. 1.2 The neck theorem The main tool we need to develop the bubble tree picture is the neck theorem which should be compared with the neck theorem in the work of Anderson and Cheeger [2]. Due to the lack of point-wise bounds on Ricci curvature, our version of the neck theorem will have weaker conclusion. But it is sufficient to allow us to construct the bubble tree at each point of curvature concentration. Let (M4, g) be a Riemannian manifold. For a point p ∈ M , denote by Br(p) the geodesic ball with radius r centered at p, Sr(p) the geodesic sphere of radius r centered at p. Consider the geodesic annulus centered at p: Ār1,r2(p) = {q ∈M : r1 ≤ dist(q, p) ≤ r2}. In general, Ār1,r2(p) may have more than one connected components. We will consider any one component Ar1,r2(p) ⊂ Ār1, r2(p) that meets the geodesic sphere of radius r2: Ar1,r2(p) ⋂ Sr2(p) 6= ∅. Let H3(Sr(p)) be the 3D-Hausdorff measure of the geodesic sphere Sr(p). 4 S.-Y.A. Chang, J. Qing and P. Yang Theorem 4. Suppose (M4, g) is a Bach flat and simply connected 4-manifold with a Yamabe metric of positive Yamabe constant. Let p ∈M , α ∈ (0, 1), ε > 0, v1 > 0, and a < dist(p, ∂M). Then there exist positive numbers δ0, c2, n depending on ε, α, Cs, v1, a such that the following holds. Let Ar1,r2(p) be a connected component of the geodesic annulus in M such that r2 ≤ c2a, r1 ≤ δ0r2, H3(Sr(p)) ≤ v1r 3, ∀ r ∈ [r1, 100r1], and ∫ Ar1,r2 (p) \|Rm\|2dv ≤ δ0. Then Ar1,r2(p) is the only such component. In addition, for the only component A (δ − 1 4 0 −ε)r1,(δ 1 4 0 +ε)r2 (p), which intersects with S (δ 1 4 0 +ε)r2 (p), there exist some Γ ⊂ O(4), acting freely on S3, with \|Γ\| ≤ n, and an quasi isometry ψ, with A (δ − 1 4 0 +ε)r1,(δ 1 4 0 −ε)r2 (p) ⊂ Ψ(C δ − 1 4 0 r1,δ 1 4 0 r2 (S3/Γ)) ⊂ A (δ − 1 4 0 −ε)r1,(δ 1 4 0 +ε)r2 (p) such that for all C 1 2 r,r(S 3/Γ) ⊂ C δ − 1 4 0 r1,δ 1 4 0 r2 (S3/Γ), in the cone coordinates, one has \|(Ψ∗(r−2g))ij − δij \|C1,α ≤ ε. The first step in the proof is to use the Sobolev inequality to show the uniqueness of the connected annulus Ar1,r2(p). The second step is to establish the growth of volume of geodesic spheres H3(Sr(p) ≤ Cr3 for all r ∈ [r1, 1 2r2]. Here we rely on the work of Tian and Viaclovsky [14, 15] where they analyzed the end structure of a Bach-flat, scalar flat manifolds with finite L2 total curvature. The last step is to use the Gromov and Cheeger compactness argument as in the work of Anderson and Cheeger [2] to get the cone structure of the neck. 1.3 Bubble tree construction In this section we attempt to give a clear picture about what happen at curvature concentration points. We will detect and extract bubbles by locating the centers and scales of curvature concentration. We will assume here that (Mi, gi) are Bach flat 4-manifolds with positive scalar curvature Yamabe metrics, vanishing first homology, and finite L2 total curvature. Choose δ small enough according to the ε-estimates and the neck theorem in the previous section. Suppose thatXi ⊂Mi contains a geodesic ball of a fixed radius r0 and∫ Tη0 (∂Xi) \|Rmi\|2dvi ≤ δ 2 , where Tη0(∂Xi) = {p ∈Mi : dist(p, ∂Xi) < η0}, Some Progress in Conformal Geometry 5 for some fixed positive number 4η0 < r0. Define, for p ∈ Xi, s1i (p) = r such that ∫ Bi r(p) \|Rmi\|2dvi = δ 2 . Let p1 i = p such that s1i (p) = inf Bi t0 (pi) s1i (p). We may assume λ1 i = s1i (p 1 i ) → 0, for otherwise there would be no curvature concentration in Xi. We then conclude that (Mi, (λ1 i ) −2gi, p 1 i ) converges to (M1 ∞, g 1 ∞, p 1 ∞), which is a Bach flat, scalar flat, complete 4-manifold satisfying the Sobolev inequality, having finite L2 total curvature, and one single end. Definition 1. We call a Bach flat, scalar flat, complete 4-manifold with the Sobolev inequality, finite L2 total curvature, and a single ALE end a leaf bubble, while we will call such space with finitely many isolated irreducible orbifold points an intermediate bubble. Now, we define, for p ∈ Xi \Bi K1λ1 i (p1 i ), s2i (p) = r such that∫ Bi r(p)\Bi K1λ1 i (p1 i ) \|Rmi\|2dvi = δ 2 . Let p2 i = p such that s2i (p) = inf Bi r(p)\Bi K1λ1 i (p1 i ) s2i (p). Again let λ2 i = s2i (p 2 i ) → 0. Otherwise there would be no more curvature concentration. Then Lemma 1. λ2 i λ1 i + dist(p1 i , p 2 i ) λ1 i →∞. There are two possibilities: Case 1. dist(p1 i , p 2 i ) λ2 i →∞; Case 2. dist(p1 i , p 2 i ) λ2 i ≤M1. In Case 1, we certainly also have dist(p1 i , p 2 i ) λ1 i →∞. Therefore, in the convergence of the sequence (Mi, (λ2 i ) −2gi, p 2 i ) the concentration which pro- duces the bubble (M1 ∞, g 1 ∞) eventually escapes to infinity of M2 ∞ and hence is not visible to the bubble (M2 ∞, g 2 ∞), likewise, in the converging sequence (Mi, (λ1 i ) −2gi, p 1 i ) one does not see the concentration which produces (M2 ∞, g 2 ∞). There are at most finite number of such leaf bubbles. 6 S.-Y.A. Chang, J. Qing and P. Yang Definition 2. We say two bubbles (M j1 ∞, g j1 ∞) and (M j2 ∞, g j2 ∞) associated with (pj1 i , λ j1 i ) and (pj2 i , λ j2 i ) are separable if dist(pj1 i , p j2 i ) λj1 i →∞ and dist(pj1 i , p j2 i ) λj2 i →∞. In Case 2, one starts to trace intermediate bubbles which will be called parents of some bubbles. We would like to emphasize a very important point here. One needs the neck theorem to take limit in Goromov–Hausdorff topology to produce the intermediate bubbles. The neck Theorem is used to prove the limit space has only isolated point singularities, which are then proven to be orbifold points. Lemma 2. Suppose that there are several separable bubbles {(M j ∞, g j ∞)}j∈J associated with {(pj i , λ j i )}j∈J . Suppose that there is a concentration detected as (pk i , λ k i ) after {(pj i , λ j i )}j∈J such that dist(pk i , p j i ) λk i ≤M j , therefore λk i λj i →∞ for each j ∈ J . In addition, suppose that {(pj i , λ j i )}j∈J is the maximal collection of such. Then (Mi, (λk i ) −2gi, p k i ) converges in Gromov–Hausdorff topology to an intermediate bubble (Mk ∞, g k ∞). (Mk ∞, g k ∞) is either a parent or a grandparent of all the given bubbles {(M j ∞, g j ∞)}j∈J . We remark that it is necessary to create some strange intermediate bubbles to handle the inseparable bubbles. This situation does not arise in the degeneration of Einstein metrics. In that case there is a gap theorem for Ricci flat complete orbifolds and there is no curvature concentration at the smooth points due to a simple volume comparison argument, both of which are not yet available in our current situation. We will call those intermediate bubbles exotic bubbles. Definition 3. A bubble tree T is defined to be a tree whose vertices are bubbles and whose edges are necks from neck Theorem. At each vertex (M j ∞, g j ∞), its ALE end is connected, via a neck, to its parent towards the root bubble of T , while at finitely many isolated possible orbifold points of (M j ∞, g j ∞), it is connected, via necks, to its children towards leaf bubbles of T . We say two bubble trees T1 and T2 are separable if their root bubbles are separable. To finish this process we just iterate the process of extracting bubbles the construction has to end at some finite steps. In summary we have Theorem 5. Suppose that (Mi, gi) are Bach flat 4-manifolds with positive scalar curvature Yamabe metrics, vanishing first homology, and finite L2 total curvature. Then (Mi, gi) converges to Bach-flat 4-manifold (M∞, g∞) with finitely orbifold singularities S. The convergence is strong in C∞ away from a finite number of points B ⊃ S. At each point b in B there is a bubble tree attached to b. Some Progress in Conformal Geometry 7 2 Conformally compact Einstein manifolds 2.1 Conformally compact Einstein manifolds Suppose that Xn+1 is a smooth manifold of dimension n+ 1 with smooth boundary ∂X = Mn. A defining function for the boundary Mn in Xn+1 is a smooth function x on X̄n+1 such that x > 0 in X; x = 0 on M ; dx 6= 0 on M. A Riemannian metric g on Xn+1 is conformally compact if (X̄n+1, x2g) is a compact Riemannian manifold with boundary Mn for a defining function x. Conformally compact manifold (Xn+1, g) carries a well-defined conformal structure on the boundary Mn, where each metric ĝ in the class is the restriction of ḡ = x2g to the boundary Mn for a defining function x. We call (Mn, [ĝ]) the conformal infinity of the conformally compact manifold (Xn+1, g). A short computation yields that, given a defining function x, Rijkl[g] = \|dx\|2ḡ(gikgjl − gilgjk) +O(x3) in a coordinate (0, ε) × Mn ⊂ Xn+1. Therefore, if we assume that g is also asymptotically hyperbolic, then \|dx\|2ḡ\|M = 1 for any defining function x. If (Xn+1, g) is a conformally compact manifold and Ric[g] = −ng, then we call (Xn+1, g) a conformally compact Einstein manifold. Given a conformally compact, asymptotically hyperbolic manifold (Xn+1, g) and a represen- tative ĝ in [ĝ] on the conformal infinity Mn, there is a uniquely determined defining function x such that, on M × (0, ε) in X, g has the normal form g = x−2(dx2 + gx), (1) where gx is a 1-parameter family of metrics on M . This is because Lemma 3. Suppose that (Xn+1, g) is a conformally compact, asymptotically hyperbolic manifold with the conformal infinity (M, [ĝ]). Then, for any ĝ ∈ [ĝ], there exists a unique defining function x such that \|dx\|2r2g = 1 in a neighborhood of the boundary [0, ε)×M and r2g\|M = ĝ. Given a conformally compact Einstein manifold (Xn+1, g), in the local product coordinates (0, ε)×Mn near the boundary where the metric takes the normal form (1), the Einstein equations split and display some similarity to a second order ordinary differential equations with a regular singular point. Lemma 4. Suppose that (Xn+1, g) is a conformally compact Einstein manifold with the con- formal infinity (Mn, [ĝ]) and that x is the defining function associated with a metric ĝ ∈ [ĝ]. Then gx = ĝ + g(2)x2 + (even powers of x) + g(n−1)xn−1 + g(n)xn + · · · , 8 S.-Y.A. Chang, J. Qing and P. Yang when n is odd, and gx = ĝ + g(2)x2 + (even powers of x) + g(n)xn + hxn log x+ · · · , when n is even, where: a) g(2i) are determined by ĝ for 2i < n; b) g(n) is traceless when n is odd; c) the trace part of g(n) is determined by ĝ and h is traceless and determined by ĝ; d) the traceless part of g(n) is divergence free. Readers are referred to [9] for more details about the above two lemmas. 2.2 Examples of conformally compact Einstein manifolds Let us look at some examples. a) The hyperbolic spaces( Rn+1, (d\|x\|)2 1 + \|x\|2 + \|x\|2dσ ) , where dσ is the standard metric on the n-sphere. We may write gH = s−2 ( ds2 + ( 1− s2 4 )2 dσ ) , where s = 2√ 1 + \|x\|2 + \|x\| is a defining function. Hence the conformal infinity is the standard round sphere (Sn, dσ). b) The hyperbolic manifolds( S1(λ)×Rn, (1 + r2)dt2 + dr2 1 + r2 + r2dσ ) . Let r = 1− s2 4 s = sinh log 2 s for a defining function s. Then g0 H = s−2 ( ds2 + ( 1 + s2 4 )2 dt2 + ( 1− s2 4 )2 dσ ) . Thus the conformal infinity is standard (S1(λ)× Sn−1, dt2 + dσ). c) AdS-Schwarzchild( R2 × S2, gm +1 ) , where gm +1 = V dt2 + V −1dr2 + r2gS2 , V = 1 + r2 − 2m r , Some Progress in Conformal Geometry 9 m is any positive number, r ∈ [rh,+∞), t ∈ S1(λ) and (θ, φ) ∈ S2, and rh is the positive root for 1 + r2 − 2m r = 0. In order for the metric to be smooth at each point where S1 collapses we need V dt2 + V −1dr2 to be smooth at r = rh, i.e. V 1 2 d(V 1 2 2πλ) dr ∣∣∣ r=rh = 2π. Note that its conformal infinity is (S1(λ) × S2, [dt2 + dθ2 + sinθ dφ2]) and S1 collapses at the totally geodesic S2, which is the so-called horizon. Interestingly, λ is does not vary monotonically in rh, while rh monotonically depends on m. In fact, for each 0 < λ < 1/ √ 3, there are two different m1 and m2 which share the same λ. Thus, for the same conformal infinity S1(λ)× S2 when 0 < λ < 1/ √ 3, there are two non-isometric AdS-Schwarzschild space with metric g+ m1 and g+ m2 on R2×S2. These are the interesting simple examples of non-uniqueness for conformally compact Einstein metrics. d) AdS-Kerr spaces( CP 2 \ {p}, gα ) , where p is a point on CP 2, gα = Eα((r2 − 1)F−1 α dr2 + (r2 − 1)−1Fα(dt+ cos θdφ)2 + (r2 − 1)(dθ2 + sin2 θdφ2)), Eα = 2 3 α− 2 α2 − 1 , Fα = (r − α)((r3 − 6r + 3α−1)Eα + 4(r − α−1)), r ≥ α, t ∈ S1(λ), and (θ, φ) ∈ S2. For the metric to be smooth at the horizon, the totally geodesic S2, we need to require√ F E(r2 − 1) d dr ( 2πλ √ EF r2 − 1 ) = 2π. Here (t, θ, φ) is the coordinates for S3 through the Hopf fiberation. The conformal infinity is the Berger sphere with the Hopf fibre of length πEα and the S2 of area 4πEα. For every 0 < λ < (2 − √ 3)/3 there are exactly two α, hence two AdS-Kerr metrics gα. It is interesting to note that (2− √ 3)/3 < 1, so the standard S3(1) is not included in this family. One may ask, given a conformal manifold (Mn, [ĝ]), is there a conformally compact Einstein manifold (Xn+1, g) such that (Mn, [ĝ]) is the conformal infinity? This in general is a difficult open problem. Graham and Lee in [10] showed that for any conformal structure that is a per- turbation of the round one on the sphere Sn there exists a conformally compact Einstein metric on the ball Bn+1. 2.3 Conformal compactifications Given a conformally compact Einstein manifold (Xn+1, g), what is a good conformal com- pactification? Let us consider the hyperbolic space. The hyperbolic space (Hn+1, gH) is the hyperboloid{ (t, x) ∈ R×Rn+1 : −t2 + \|x\|2 = −1, t > 0 } in the Minkowski space-time R1,n+1. The stereographic projection via the imaginary south pole gives the Poincaré ball model( Bn+1, ( 2 1− \|y\|2 )2 \|dy\|2 ) 10 S.-Y.A. Chang, J. Qing and P. Yang and replacing the x-hyperplane by z-hyperplane tangent to the light cone gives the half-space model( Rn+1 + , \|dz\|2 z2 n+1 ) , where 1 + \|y\|2 1− \|y\|2 = t, 1 zn+1 = t− xn+1. Therefore (Hn+1, t−2gH) = (Sn+1 + , gSn+1), (Hn+1, (t+ 1)−2gH) = (Bn+1, \|dy\|2), (Hn+1, (t− xn+1)−2gH) = (Rn+1 + , \|dz\|2). The interesting fact here is that all coordinate functions {t, x1, x2, . . . , xn+1} of the Minkowski space-time are eigenfunctions on the hyperboloid. Thus positive eigenfunctions on a conformally compact Einstein manifold are expected to be candidates for good conformal compactifications. This is first observed in [12]. Lemma 5. Suppose that (Xn+1, g) is a conformally compact Einstein manifold and that x is a special defining function associated with a representative ĝ ∈ [ĝ]. Then there always exists a unique positive eigenfunction u such that ∆u = (n+ 1)u in X and u = 1 x + R[ĝ] 4n(n− 1) x+O(x2) near the infinity. We remark here that, for the hyperbolic space Hn+1 and the standard round metric in the infinity, we have t = 1 x + 1 4 x. As we expect, positive eigenfunctions indeed give a preferable conformal compactification. Theorem 6. Suppose that (Xn+1, g) is a conformally compact Einstein manifold, and that u is the eigenfunction obtained for a Yamabe metric ĝ of the conformal infinity (M, [ĝ]) in the previous lemma. Then (Xn+1, u−2g) is a compact manifold with totally geodesic boundary M and R[u−2g] ≥ n+ 1 n− 1 R[ĝ]. As a consequence Corollary 1. Suppose that (Xn+1, g) is a conformally compact Einstein manifold and its confor- mal infinity is of positive Yamabe constant. Suppose that u is the positive eigenfunction associa- ted with the Yamabe metric on the conformal infinity obtained in Lemma 1. Then (Xn+1, u−2g) is a compact manifold with positive scalar curvature and totally geodesic boundary. Some Progress in Conformal Geometry 11 The work of Schoen–Yau and Gromov–Lawson then give some topological obstruction for a conformally compact Einstein manifold to have its conformal infinity of positive Yamabe constant. A surprising consequence of the eigenfunction compactifications is the rigidity of the hyperbolic space without assuming the spin structure. Theorem 7. Suppose that (Xn+1, g) is a conformally compact Einstein manifold with the round sphere as its conformal infinity. Then (Xn+1, g) is isometric to the hyperbolic space. 2.4 Renormalized volume We will introduce the renormalized volume, which was first noticed by physicists in their inves- tigations of the holography principles in AdS/CFT. Take a defining function x associated with a choice of the metric ĝ ∈ [ĝ] on the conformal infinity, then compute, when n is odd, Vol({x > ε} = c0ε −n + odd powers of ε+ V + o(1), (2) when n is even, Vol({s > ε}) = c0ε −n + even powers of ε+ L log 1 ε + V + o(1). (3) It turns out the numbers V in odd dimension and L in even dimension are independent of the choice of the metrics in the class. We will see that V in even dimension is in fact a conformal anomaly. Lemma 6. Suppose that (Xn+1, g) is a conformally compact Einstein manifold and that x̄ and x are two defining functions associated with two representatives in [ĝ] on the conformal infinity (Mn, [ĝ]). Then x̄ = xew for a function w on a neighborhood of the boundary [0, ε)×M whose expansion at x = 0 consists of only even powers of x up through and including xn+1 term. Theorem 8. Suppose that (Xn+1, g) is a conformally compact Einstein manifold. The V in (2) when n is odd and L in (3) when nn is even are independent of the choice of representative ĝ ∈ [ĝ] on the conformal infinity (Mn, [ĝ]). Let us calculate the renormalized volume for the examples in Section 2.2. a) The hyperbolic space: We recall (H4, gH) = ( B4, ( 2 1− \|y\|2 )2 \|dy\|2 ) , where gH = s−2 ( ds2 + ( 1− s2 4 )2 h0 ) and h0 is the round metric on S3. Then vol({s > ε}) = ∫ 2 ε ∫ S3 s−4 ( 1− s2 4 )3 dσ0ds 12 S.-Y.A. Chang, J. Qing and P. Yang where dσ0 is the volume element for the round unit sphere vol({s > ε}) = 2π2 ∫ 2 ε s−4 ( 1− 3s2 4 + 3s4 16 − s6 64 ) ds = 2π2 ( −1 3 s−3 ∣∣∣2 ε + 3 4 s−1 ∣∣∣2 ε + 3 16 (2− ε)− 1 3× 64 s3 ∣∣∣2 ε ) = 2π2 3 ε−3 − 3π2 2 ε−1 + 2π2 ( − 1 3× 8 + 3 8 + 3 8 − 1 3× 8 ) +O(ε) = 2π2 3 ε−3 − 3π2 2 ε−1 + 4π2 3 +O(ε). Thus V (H4, gH) = 4π2 3 . b) The hyperbolic manifold: We recall( S1(λ)×R3, (1 + r2)dt2 + dr2 1 + r2 + r2gS2 ) and g0 H = s−2 ( ds2 + ( 1− s2 4 )2 (dθ2 + sin2 θdφ2) + ( 1 + s2 4 )2 dt2 ) . Then vol({s > ε}) = ∫ 2 ε ∫ S2 ∫ S1 s−4 ( 1− s2 4 )2( 1 + s2 4 ) dω0dtds where dω0 stands for the volume element for the round unit sphere S2 vol({s > ε}) = 8π2 ∫ 2 ε s−4 ( 1− s2 2 + s4 16 )( 1 + s2 4 ) ds = 8π2λ ∫ 2 ε s−4 ( 1− s2 4 − s4 16 + s6 64 ) ds = 8π2 3 λε−3 − 2π2λε−1 + 8π2λ ( − 1 3× 8 + 1 8 − 1 8 + 1 3× 8 ) +O(ε) = 8π2 3 λε−3 − 2π2λε−1 +O(ε). Thus V (S1 ×R3, g0 H) = 0. c) AdS-Schwarzschild spaces: We recall on S2 ×R2 gm +1 = ( 1 + r2 − 2m r ) dt2 + dr2 1 + r2 − 2m r + r2(dθ2 + sin2 θdφ2). First let us find the special defining function, i.e. to have 1 1 + r2 − 2m r dr2 = s−2ds2 Some Progress in Conformal Geometry 13 that is, if denote by r = ρ/s, where ρ = ρ(s), ρ− sρ′ = √ ρ2 + s2 − 2ms3/ρ, and ρ(0) = 1. One may solve it in power series ρ = 1− 1 4 s2 + m 3 s3 + · · · . Then gm +1 = s−2 ( ds2 + ( ρ2 + s2 − 2ms3 ρ ) dt2 + ρ2(dθ2 + sin2 θdφ2) ) . Note that s ∈ [ε, sh] for r ∈ [rh,Mε], log sh = log ε+ ∫ Mε rh 1√ 1 + r2 − 2m r dr < +∞, and Mε = ε−1ρ(ε) = ε−1 ( 1− 1 4 ε2 + m 3 s3 + · · · ) . Therefore vol({s > ε}) = ∫ sh ε ∫ S1(λ) ∫ S2 s−4 √ ρ2 + s2 − 2ms2 ρ ρ2dtdσ0ds = 8π2λ ∫ sh ε s−4 √ ρ2 + s2 − 2ms3 ρ ρ2ds = 8π2λ ∫ M rh s−1 √ 1 + r2 − 2m r r2 ( −ds dr ) dr = 8π2λ ∫ M rh r2dr = 8π2λ 3 (M3 − r3h). Thus the renormalized volume V (R2 × S2, gm +1) = 8π2 3 r2h(1− r2h) 3r2h + 1 , where V (R2 × S2, gm +1) < 0 when rh > 1; V (R2 × S2, gm +1) = 0 only when rh = 1 or 0; and it achieves its maximum value at rh = 1/ √ 3 V (R2 × S2, gm +1)max = 1 9 · 4π2 3 χ(R2 × S2). d) AdS-Kerr spaces: We will omit the calculation here. The renormalized volume V (CP2 \ {p}, gα) = 4π2Eα ( −1 6 Eα(α3 + 3α−1) + 2 3 (α+ α−1) ) . Clearly, V (CP2 \{p}, gα) goes to zero when α goes to 2, and V (CP2 \{p}, gα) goes to −∞ when α goes to ∞. One may find the maximum value for the renormalized volume is achieved at α = 2 + √ 3. Therefore V (CP2 \ {p}, gα)max = 4π2 3 · 2(4− √ 3) 9 < 1 2 · 4π2 3 χ(CP2 \ {p}). 14 S.-Y.A. Chang, J. Qing and P. Yang 2.5 Renormalized volume and Chern–Gauss–Bonnet formula We start with the Gauss–Bonnet formula on a surface (M2, g) 4πχ(M2) = ∫ M Kdvg, where K is the Gaussian curvature of (M2, g). The transformation of the Gaussian curvature under a conformal change of metrics gw = e2wg is governed by the Laplacian as follows: −∆gw +K[g] = K[e2wg]e2w. The Gauss–Bonnet formula for a compact surface with boundary (M2g) is 4πχ(M) = ∫ M KdVg + 2 ∫ ∂M kdσg, where k is the geodesic curvature for ∂M in (M, g). The transformation of the geodesic curvature under a conformal change of metric gw = e2wg is −∂nw + k[g] = k[e2wg]ew, where ∂n is the inward normal derivative. Notice that −∆[e2wg] = e−2w(−∆[g]), −∂n[e2wg] = e−w(−∂n[g]), for which we say they are conformally covariant. In four dimension there is a rather complete analogue. We may write the Chern–Gauss–Bonnet formula in the form 8π2χ(M4) = ∫ M (\|W \|2 +Q)dVg for closed 4-manifold and 8π2χ(M4) = ∫ M (\|W \|2 +Q)dVg + 2 ∫ ∂M (L+ T )dσg, where W is the Weyl curvature, L is a point-wise conformal invariant curvature of ∂M in (M, g). Q = 1 6 (R2 − 3\|Ric\|2 −∆R), T = − 1 12 ∂nR+ 1 6 RH −RαnβnLαβ + 1 9 H3 − 1 3 TrL3 − 1 3 ∆̃H, R is the scalar curvature, Ric is the Ricci curvature, L is the second fundamental form of ∂M in (M, g). We know the transformation of Q under a conformal change metric gw = e2wg is P4[g]w +Q[g] = Q[e2wg]e4w, where P4 = (−∆)2 + δ { 2 3 Rg − 2Ric } d is the so-called Paneitz operator, and the transformation of T is P3[g]w + T [g] = T [e2w]e3w, Some Progress in Conformal Geometry 15 where P3 = 1 2 ∂n∆g − ∆̃∂n + 2 3 H∆̃ + Lαβ∇̃α∇̃β + 1 3 ∇̃αH · ∇̃α + ( F − 1 3 R ) ∂n. We also have P4[e2wg] = e−4wP4[g], P3[e2wg] = e−3wP3[g]. On the other hand, to calculate the renormalized volume in general, for odd n, upon a choice of a special defining function x, one may solve −∆v = n in Xn+1 for v = log x+A+Bxn, A, B are even in x, and A\|x=0 = 0. Let Bn[g, ĝ] = B\|x=0. Fefferman and Graham observed Lemma 7. V (Xn+1, g) = ∫ M Bn[g, ĝ]dv[ĝ]. We observe that the function v in the above is also good in conformal compactifications. For example, given a conformally compact Einstein 4-manifold (X4, g), let us consider the com- pactification (X4, e2vg). Then Q4[e2vg] = 0 and its boundary is totally geodesic in (X4, e2vg). Moreover T [e2vg] = 3B3[g, ĝ]. Therefore we obtain easily the following generalized Chern–Gauss–Bonnet formula. Proposition 1. Suppose that (X4, g) is a conformally compact Einstein manifold. Then 8π2χ(X4) = ∫ X4 (\|W \|2dv)[g] + 6V (X4, g). 2.6 Topology of conformally compact Einstein 4-manifolds In the following let us summarize some of our works appeared in [6]. From the generalized Chern–Gauss–Bonnet formula, obviously V ≤ 4π2 3 χ(X) and the equality holds if and only if (X4, g) is hyperbolic. Comparing with Chern–Gauss–Bonnet formula for a closed 4-manifold 1 8π2 ∫ M4 (\|W \|2 + σ2)dv = χ(M4) 16 S.-Y.A. Chang, J. Qing and P. Yang one sees that the renormalized volume replaces the role of the integral of σ2. In the following we will report some results on the topology of a conformally compact Einstein 4-manifold in terms of the size of the renormalized volume relative to the Euler number, which is analogous to the results of Chang–Gursky–Yang [7, 8] on a closed 4-manifold with positive scalar curvature and large integral of σ2 relative to the Euler number. The proofs mainly rely on the conformal compactifications discussed earlier, a simple doubling argument and applications of the above mentioned results of Chang–Gursky–Yang [7, 8]. Theorem 9. Suppose (X4, g) is a conformally compact Einstein 4-manifold with its confor- mal infinity of positive Yamabe constant and the renormalized volume V is positive. Then H1(X,R) = 0. Theorem 10. Suppose (X4, g) is a conformally compact Einstein 4-manifold with conformal infinity of positive Yamabe constant. Then V > 1 3 4π2 3 χ(X) implies that H2(X,R) vanishes. A nice way to illustrate the above argument is the following. We may consider the modified Yamabe constant Y λ(M, [g]) = inf g∈[g] ∫ M (R[g] + λ\|W+\|g)dvg ( ∫ M dvg) n−2 n . Then, one knows that (M, [g]) is of positive Y λ(M, [g]) if and only if there is a metric g ∈ [g] with R+ λ\|W+\| > 0. As a consequence of the following Bochner formula ∆ 1 2 \|ω\|2 = \|∇ω\|2 − 2W+(ω, ω) + 1 3 R\|ω\|2 ≥ \|∇ω\|2 + (R− 2 √ 6\|W+\|)\|ω\|2 for any self-dual harmonic 2-form ω, one easily sees that a closed oriented 4-manifold with Y −2 √ 6 > 0 has its b+2 = 0. We also observe Theorem 11. Suppose (X4, g) is a conformally compact Einstein 4-manifold with its conformal infinity of positive Yamabe constant and that V > 1 2 4π2 3 χ(X). Then X is diffeomorphic to B4 and more interestingly M is diffeomorphic to S3. The detailed proofs of the above theorems are in our paper [6]. One may recall a) V (H4, gH) = 4π2 3 , b) V (S1 ×R3, gH) = 0, c) V (S2 ×R2, gm + ) = 8π2 3 r2h(1− r2h) 3r2h + 1 ≤ 1 9 4π2 3 χ(S2 ×R2), d) V (CP 2 \B, gK) ≤ 4π2 3 2(4− √ 3) 9 < 1 3 4π2 3 χ(CP 2 \B). Theorem 10 is rather sharp, in cases (c) and (d) the second homology is nontrivial while the renormalized volume is very close to one-third of the maximum. Some Progress in Conformal Geometry 17 References [1] Anderson M., Orbifold compactness for spaces of Riemannian metrics and applications, Math. Ann. 331 (2005), 739–778, math.DG/0312111. [2] Anderson M., Cheeger J., Diffeomorphism finiteness for manifolds with Ricci curvature and Ln/2-norm of curvature bounded, Geom. Funct. Anal. 1 (1991), 231–252. [3] Brezis H., Coron J.M., Convergence of solutions of H-systems or how to blow bubbles, Arch. Ration. Mech. Anal. 89 (1985), 21–56. [4] Bray H., Neves A., Classification of prime 3-manifolds with Yamabe invariant larger than RP 3, Ann. of Math. (2) 159 (2004), 407–424. [5] Chang S.-Y.A., Qing J., Yang P., On a conformal gap and finiteness theorem for a class of four-manifolds. Geom. Funct. Anal. 17 (2007), 404–434, math.DG/0508621. [6] Chang S.-Y.A., Qing J., Yang P., On the topology of conformally compact Einstein 4-manifolds, in Non- compact Problems at the Intersection of Geometry, Analysis, and Topology, Contemp. Math. 350 (2004), 49–61, math.DG/0305085. [7] Chang S.-Y.A., Gursky M., Yang P., An equation of Monge–Ampére type in conformal geometry and 4- manifolds of positive Ricci curvature, Ann. of Math. (2) 155 (2002), 709–787, math.DG/0409583. [8] Chang S.-Y.A., Gursky M., Yang P., An apriori estimate for a fully nonlinear equation on 4-manifolds, J. D’Analyse Math. 87 (2002), 151–186. [9] Graham C.R., Volume and area renormalizations for conformally compact Einstein metrics, in The Pro- ceedings of the 19th Winter School “Geometry and Physics” (1999, Srǹı), Rend. Circ. Mat. Palermo (2) 63 (2000), suppl., 31–42, math.DG/9909042. [10] Graham C.R., Lee J., Einstein metrics with prescribed conformal infinity on the ball, Adv. Math. 87 (1991), 186–225. [11] Qing J., On singularities of the heat flow for harmonic maps from surfaces into spheres, Comm. Anal. Geom. 3 (1995), 297–315. [12] Qing J., On the rigidity for conformally compact Einstein manifolds, Int. Math. Res. Not. 2003 (2003), 1141–1153, math.DG/0305084. [13] Struwe M., Global compactness result for elliptic boundary value problem involving limiting nonlinearities, Math. Z. 187 (1984), 511–517. [14] Tian G., Viaclovsky J., Bach flat asymptotically ALE metrics, Invent. Math. 160 (2005), 357–415, math.DG/0310302. [15] Tian G., Viaclovsky J., Moduli space of critical Riemannian metrics in dimension 4, Adv. Math. 196 (2005), 346–372, math.DG/0312318. http://arxiv.org/abs/math.DG/0312111 http://arxiv.org/abs/math.DG/0508621 http://arxiv.org/abs/math.DG/0305085 http://arxiv.org/abs/math.DG/0409583 http://arxiv.org/abs/math.DG/9909042 http://arxiv.org/abs/math.DG/0305084 http://arxiv.org/abs/math.DG/0310302 http://arxiv.org/abs/math.DG/0312318 1 Conformal gap and finiteness theorem for a class of closed 4-manifolds 1.1 Introduction 1.2 The neck theorem 1.3 Bubble tree construction 2 Conformally compact Einstein manifolds 2.1 Conformally compact Einstein manifolds 2.2 Examples of conformally compact Einstein manifolds 2.3 Conformal compactifications 2.4 Renormalized volume 2.5 Renormalized volume and Chern-Gauss-Bonnet formula 2.6 Topology of conformally compact Einstein 4-manifolds References

Some Progress in Conformal Geometry

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