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 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 $\sigma_2$-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.

1 Conformal gap and finiteness theorem for a class of closed 4-manifolds 1

.1 Introduction
Suppose that (M 4 , g) is a closed 4-manifold. It follows from the positive mass theorem that, for a 4-manifold with positive Yamabe constant, M σ 2 dv ≤ 16π 2 and equality holds if and only if (M 4 , g) is conformally equivalent to the standard 4-sphere, where 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.
One may ask, whether there is a constant ǫ 0 > 0 such that a closed 4-manifold M 4 has to be diffeomorphic to S 4 if it admits a metric g with positive Yamabe constant and M σ 2 [g]dv g ≥ (1 − ǫ)16π 2 .
for some ǫ < ǫ 0 ? 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 σ 2 . We will answer the question affirmatively in the class of Bach flat 4-manifolds.
Recall that Riemann curvature tensor decomposes into in dimension 4, where W ijkl is the Weyl curvature, Rg ij is Weyl-Schouten curvature tensor and R ij is the Ricci curvature tensor. Also recall that the Bach tensor is We say that a metric g is Bach flat if B ij = 0. Bach flat metrics are critical metrics for the functional M |W | 2 dv. Bach flatness is conformally invariant in dimension 4. It follows from Chern-Gauss-Bonnet, Our approach is based on the recent work on the compactness of Bach flat metrics on 4manifolds 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 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: for some fixed positive number Λ 0 , and for some fixed positive number σ 0 . 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 g = e 2w g such that σ 2 (Aḡ) = 1, which we shall call the σ 2 metric. The bubble tree structure in the degeneration of Yamabe metrics in A is also helpful to understand the behavior of the σ 2 -metrics in A. For example: Theorem 3. For the conformal classes [g 0 ] ∈ A the conformal metrics g = e 2w g 0 satisfying the equation σ 2 (g) = 1 has a uniform bound for the diameter.
The detailed version of this work has appeared in our paper [5].

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 (M 4 , g) be a Riemannian manifold. For a point p ∈ M , denote by B r (p) the geodesic ball with radius r centered at p, S r (p) the geodesic sphere of radius r centered at p. Consider the geodesic annulus centered at p: In general,Ā r 1 ,r 2 (p) may have more than one connected components. We will consider any one component A r 1 ,r 2 (p) ⊂Ār 1 , r 2 (p) that meets the geodesic sphere of radius r 2 : Let H 3 (S r (p)) be the 3D-Hausdorff measure of the geodesic sphere S r (p). Then there exist positive numbers δ 0 , c 2 , n depending on ǫ, α, C s , v 1 , a such that the following holds. Let A r 1 ,r 2 (p) be a connected component of the geodesic annulus in M such that and Ar 1 ,r 2 (p) Then A r 1 ,r 2 (p) is the only such component. In addition, for the only component and an quasi isometry ψ, with The first step in the proof is to use the Sobolev inequality to show the uniqueness of the connected annulus A r 1 ,r 2 (p). The second step is to establish the growth of volume of geodesic spheres 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 L 2 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.

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 (M i , g i ) are Bach flat 4-manifolds with positive scalar curvature Yamabe metrics, vanishing first homology, and finite L 2 total curvature. Choose δ small enough according to the ǫ-estimates and the neck theorem in the previous section. Suppose that X i ⊂ M i contains a geodesic ball of a fixed radius r 0 and for some fixed positive number 4η 0 < r 0 . Define, for p ∈ X i , Let We may assume , which is a Bach flat, scalar flat, complete 4-manifold satisfying the Sobolev inequality, having finite L 2 total curvature, and one single end. Definition 1. We call a Bach flat, scalar flat, complete 4-manifold with the Sobolev inequality, finite L 2 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. Let Otherwise there would be no more curvature concentration. Then Lemma 1.
There are two possibilities: In Case 1, we certainly also have Therefore, in the convergence of the sequence (M i , (λ 2 i ) −2 g i , p 2 i ) the concentration which produces the bubble (M 1 ∞ , g 1 ∞ ) eventually escapes to infinity of M 2 ∞ and hence is not visible to the bubble (M 2 ∞ , g 2 ∞ ), likewise, in the converging sequence . There are at most finite number of such leaf bubbles.
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.
is either a parent or a grandparent of all the given bubbles 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 T 1 and T 2 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

Conformally compact Einstein manifolds
Suppose that X n+1 is a smooth manifold of dimension n + 1 with smooth boundary ∂X = M n . A defining function for the boundary M n in X n+1 is a smooth function x onX n+1 such that A Riemannian metric g on X n+1 is conformally compact if (X n+1 , x 2 g) is a compact Riemannian manifold with boundary M n for a defining function x. Conformally compact manifold (X n+1 , g) carries a well-defined conformal structure on the boundary M n , where each metricĝ in the class is the restriction ofḡ = x 2 g to the boundary M n for a defining function x. We call (M n , [ĝ]) the conformal infinity of the conformally compact manifold (X n+1 , g). A short computation yields that, given a defining function x, in a coordinate (0, ǫ) × M n ⊂ X n+1 . Therefore, if we assume that g is also asymptotically hyperbolic, then for any defining function x. If (X n+1 , g) is a conformally compact manifold and Ric[g] = −ng, then we call (X n+1 , g) a conformally compact Einstein manifold. Given a conformally compact, asymptotically hyperbolic manifold (X n+1 , g) and a representativeĝ in [ĝ] on the conformal infinity M n , there is a uniquely determined defining function x such that, on M × (0, ǫ) in X, g has the normal form where g x is a 1-parameter family of metrics on M . This is because Given a conformally compact Einstein manifold (X n+1 , g), in the local product coordinates (0, ǫ)×M n 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 (X n+1 , g) is a conformally compact Einstein manifold with the conformal infinity (M n , [ĝ]) and that x is the defining function associated with a metricĝ ∈ [ĝ]. Then g x =ĝ + g (2) x 2 + (even powers of x) + g (n−1) x n−1 + g (n) x n + · · · , when n is odd, and g x =ĝ + g (2) x 2 + (even powers of x) + g (n) x n + hx n log x + · · · , when n is even, where: a) g (2i) are determined byĝ for 2i < n; b) g (n) is traceless when n is odd; c) the trace part of g (n) is determined byĝ and h is traceless and determined byĝ; d) the traceless part of g (n) is divergence free.
Readers are referred to [9] for more details about the above two lemmas.

Examples of conformally compact Einstein manifolds
Let us look at some examples.
a) The hyperbolic spaces where dσ is the standard metric on the n-sphere. We may write where s = 2 1 + |x| 2 + |x| is a defining function. Hence the conformal infinity is the standard round sphere (S n , dσ).

c) AdS-Schwarzchild
m is any positive number, r ∈ [r h , +∞), t ∈ S 1 (λ) and (θ, φ) ∈ S 2 , and r h is the positive root for 1 + r 2 − 2m r = 0. In order for the metric to be smooth at each point where S 1 collapses we need V dt 2 + V −1 dr 2 to be smooth at r = r h , i.e. Note that its conformal infinity is (S 1 (λ) × S 2 , [dt 2 + dθ 2 + sin θ dφ 2 ]) and S 1 collapses at the totally geodesic S 2 , which is the so-called horizon. Interestingly, λ is does not vary monotonically in r h , while r h monotonically depends on m. In fact, for each 0 < λ < 1/ √ 3, there are two different m 1 and m 2 which share the same λ. Thus, for the same conformal infinity S 1 (λ) × S 2 when 0 < λ < 1/ √ 3, there are two non-isometric AdS-Schwarzschild space with metric g + m 1 and g + m 2 on R 2 ×S 2 . These are the interesting simple examples of non-uniqueness for conformally compact Einstein metrics.

d) AdS-Kerr spaces
where p is a point on CP 2 , r ≥ α, t ∈ S 1 (λ), and (θ, φ) ∈ S 2 . For the metric to be smooth at the horizon, the totally geodesic S 2 , we need to require Here (t, θ, φ) is the coordinates for S 3 through the Hopf fiberation. The conformal infinity is the Berger sphere with the Hopf fibre of length πE α and the S 2 of area 4πE α . For every 0 < λ < (2 − √ 3)/3 there are exactly two α, hence two AdS-Kerr metrics g α . It is interesting to note that (2 − √ 3)/3 < 1, so the standard S 3 (1) is not included in this family. One may ask, given a conformal manifold (M n , [ĝ]), is there a conformally compact Einstein manifold (X n+1 , g) such that (M n , [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 perturbation of the round one on the sphere S n there exists a conformally compact Einstein metric on the ball B n+1 .

Conformal compactifications
Given a conformally compact Einstein manifold (X n+1 , g), what is a good conformal compactification? Let us consider the hyperbolic space. The hyperbolic space (H n+1 , g H ) is the hyperboloid in the Minkowski space-time R 1,n+1 . The stereographic projection via the imaginary south pole gives the Poincaré ball model and replacing the x-hyperplane by z-hyperplane tangent to the light cone gives the half-space model The interesting fact here is that all coordinate functions {t, x 1 , x 2 , . . . , x n+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 (X n+1 , g) is a conformally compact Einstein manifold and that x is a special defining function associated with a representativeĝ ∈ [ĝ]. Then there always exists a unique positive eigenfunction u such that near the infinity.
We remark here that, for the hyperbolic space H n+1 and the standard round metric in the infinity, we have As we expect, positive eigenfunctions indeed give a preferable conformal compactification.
As a consequence Corollary 1. Suppose that (X n+1 , g) is a conformally compact Einstein manifold and its conformal infinity is of positive Yamabe constant. Suppose that u is the positive eigenfunction associated with the Yamabe metric on the conformal infinity obtained in Lemma 1. Then (X n+1 , u −2 g) is a compact manifold with positive scalar curvature and totally geodesic boundary.
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 (X n+1 , g) is a conformally compact Einstein manifold with the round sphere as its conformal infinity. Then (X n+1 , g) is isometric to the hyperbolic space.

Renormalized volume
We will introduce the renormalized volume, which was first noticed by physicists in their investigations of the holography principles in AdS/CFT. Take a defining function x associated with a choice of the metricĝ ∈ [ĝ] on the conformal infinity, then compute, when n is odd, when n is even, 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.
Thus the renormalized volume = 0 only when r h = 1 or 0; and it achieves its maximum value at

d) AdS-Kerr spaces:
We will omit the calculation here. The renormalized volume Clearly, V (CP 2 \ {p}, g α ) goes to zero when α goes to 2, and V (CP 2 \ {p}, g α ) goes to −∞ when α goes to ∞. One may find the maximum value for the renormalized volume is achieved at α = 2 + √ 3. Therefore

Renormalized volume and Chern-Gauss-Bonnet formula
We start with the Gauss-Bonnet formula on a surface (M 2 , g) where K is the Gaussian curvature of (M 2 , g). The transformation of the Gaussian curvature under a conformal change of metrics g w = e 2w g is governed by the Laplacian as follows: The Gauss-Bonnet formula for a compact surface with boundary (M 2 g) is where k is the geodesic curvature for ∂M in (M, g). The transformation of the geodesic curvature under a conformal change of metric g w = e 2w g is where ∂ n is the inward normal derivative. Notice that 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 where W is the Weyl curvature, L is a point-wise conformal invariant curvature of ∂M in (M, g).
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 g w = e 2w g is where is the so-called Paneitz operator, and the transformation of T is where We also have 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 A, B are even in x, and A| x=0 = 0. Let B n [g,ĝ] = B| x=0 .
We observe that the function v in the above is also good in conformal compactifications. For example, given a conformally compact Einstein 4-manifold (X 4 , g), let us consider the compactification (X 4 , e 2v g). Then Therefore we obtain easily the following generalized Chern-Gauss-Bonnet formula.
Proposition 1. Suppose that (X 4 , g) is a conformally compact Einstein manifold. Then

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 one sees that the renormalized volume replaces the role of the integral of σ 2 . 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 σ 2 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]. implies that H 2 (X, R) vanishes.
The detailed proofs of the above theorems are in our paper [6]. One may recall 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.