Perturbation compactness and uniqueness for a class of conformally compact Einstein manifolds

1.1 Introduction

Let X be a smooth manifold of dimension d with d ≥ 3 with boundary ∂X. A smooth conformally compact metric g ⁺ on X is a Riemannian metric such that g = r ² g ⁺ extends smoothly to the closure X ̄ for some defining function r. A defining function r is a smooth nonnegative function on the closure X ̄ such that ∂X = {r = 0} and the differential Dr ≠ 0 on ∂X. A conformally compact metric g ⁺ on X is said to be conformally compact Einstein (CCE) if, in addition,

The most significant feature of CCE manifolds (X, g ⁺) is that the metric g ⁺ canonically determines the conformal structure [ g ̂ ] on the boundary ∂X, where g ̂ = g | T ∂ X . ( ∂ X , [ g ̂ ] ) is called the conformal infinity of the conformally compact manifold (X, g ⁺). It is of great interest in both the mathematics and theoretic physics communities to understand the correspondences between conformally compact Einstein manifolds (X, g ⁺) and their conformal infinities ( ∂ X , [ g ̂ ] ) , partially due to the interest of AdS/CFT correspondence in theoretic physics (cf. Maldacena [1]–[3] and Witten [4]).

The project we work on in this paper is to address the compactness issue: Given a sequence of CCE manifolds X , g i + with M = ∂X and { g i } = r i 2 g i + a sequence of compactified metrics, with h _i = g _i | _M ; assuming {h _i } forms a compact family of metrics in M, when is it true that some representatives g ̄ i ∈ [ g i ] with { g ̄ i | M = h i } also forms a compact family of metrics in X?

We remark that, for a CCE manifold, given any conformal infinity (M, h), a special defining function r, which we call the geodesic defining function, exists so that | ∇ r 2 g + r | ≡ 1 in an asymptotic neighborhood M × [0, ϵ) of M with r ² g ⁺| _M = h. We also remark that the eventual goal in the study of the compactness problem is to establish the existence result of conformal fill-ins for some classes of Riemannian manifolds as the conformal infinities.

One of the difficulties to address the compactness problem is due to the existence of a non-local term in the asymptotic expansion of the metric near the conformal infinity. To see this, we recall the asymptotic behavior of the compactified metric g of CCE manifold (X, g ⁺) of dimension d, with conformal infinity (M, [h]), which has been worked out earlier (see Refs. [5], [6]).^[1] It turns out that the behavior is a bit different depending on whether the dimension d is even or odd.

When d is even, we have the expansion:

on an asymptotic neighborhood of M × (0, ϵ), where r denotes the geodesic defining function corresponding to the conformal infinity (∂X, h). The g ^(j) are tensors on M, and g ^(d−1) is trace-free with respect to the metric h. For j even and 0 ≤ j ≤ d − 2, the tensor g ^(j) is locally formally determined by the conformal representative h, but g ^(d−1) is a non-local term which is not determined by h, subject to the trace free condition.

When d is odd, the analogous expansion is

where the g ^(j) terms are locally determined for j even and 0 ≤ j ≤ d − 2, k is locally determined and trace-free, the trace of g ^(d−1) is locally determined, but the trace-free part of g ^(d−1) is again not determined by h. We remark that h together with g ^(d−1) determine the whole asymptotic behavior of g ([5], [10]) near the conformal infinity.

A model case of a CCE manifold is the hyperbolic ball B d with the Poincaré metric g _H with the conformal infinity the standard metric h _c on the unit d − 1 sphere S d − 1 . In this case, it was proved by Ref. [11] (see also Ref. [12] and later on by Ref. [13]) that B d , g H is the unique CCE manifold with metric h _c on S d − 1 as its conformal infinity.

Another class of examples of CCE manifolds was constructed by Graham–Lee [14], where they proved that any metric on S d − 1 close enough in the C ^2,α norm to h _c is the conformal infinity of some CCE metric on the Euclidean unit ball B d for all d ≥ 4.

In an earlier paper [15], in the special case when the dimension d = 4, we have established a compactness result for classes of CCE manifolds and derived as a consequence the uniqueness of the CCE extensions of Graham and Lee for the class of metrics on S 3 which are C ^3,α close to h _c on S 3 .

The goal of this paper to extend the results in Ref. [15] to all dimensions d ≥ 4.

Recall that when d = 4, in Refs. [15], [16], we have considered a special choice of compactified metric g* = e ^2w g ⁺ defined on a CCE manifold (X ⁴, g ⁺) of dimension four; which we named as Fefferman–Graham (FG) compactification. This metric is defined by solving the PDE [17], Theorem 4.1:

where (w − log r)|_∂X = 0.

On a general d-dimensional CCE manifold (X, g ⁺), when d > 4, we will consider a choice of the compactified metric g* which is a special case of a general class of metrics named as “adapted metrics” in an earlier paper by Case-Chang [18], Section 6. The metric was defined by solving the Poisson equation

with the Dirichlet data the constant function one, g * : = v 4 d − 4 g + with g*| _M = h, some fixed metric on the conformal infinity of (X, g ⁺). It is known that g* has free Q-curvature (see Refs. [18], [19], see also the discussion in Lemma 2.2 in the current paper).

In this paper, we first consider the case when d is even. In this case, it turns out the method of proof in Ref. [15] for d = 4 case can be directly generalized. A key property we will use is the existence of the obstruction tensor [5], [20] when d is even and which vanishes for metrics conformal to Einstein metrics. When d = 4, this obstruction tensor is the Bach tensor, and for metrics conformal to Einstein metrics, the Bach tensor vanishes (i.e. they are Bach flat). As we will see in the proof of Theorem 1.1 in Section 3 below, the equation of obstruction flat tensor is an elliptic equation which allows us to derive an ɛ-regularity property for the compactified metrics g* (under the assumptions of Theorem 1.1), and this in turn allows us to gain the regularity of the metric. This gain is the key step which allows us to apply a contradiction argument to reach the compactness result in the statement of Theorem 1.1 below.

When the dimension d of the manifold X is odd, in general, we would not expect the strong estimate C ^d−1 as in the cases when d is even due to the kr ^d−1 log r term in the expansion of the metric g in 1.2. The coefficient k of this term happens to be the obstruction tensor [5], [20] defined on the boundary of X and which in general may not vanish. Thus when d is odd, we will apply a different strategy to gain the regularity of the compactified metric g*. It turns out this strategy actually works for all dimensions d under the somewhat stronger regularity assumption C ⁶ on the boundary metrics when d is small. Instead of exploring the property of vanishing of the obstruction tensor of the metric as in the case when d is even, we will explore the regularity property of its associated Einstein metric g ⁺. In order to do so, in Section 4 below, we will modify the gauge fixing techniques developed earlier in the works [14], [21]–[23] for Einstein metrics. We first obtain the regularity of the adapted metrics near the neighborhood of the conformal infinity; we next introduce some suitable weighted spaces and apply the functional analytic techniques for such spaces to avoid the degeneracy and obtain the ɛ regularity of the adapted metric g*.

The analysis in Sections 3 and 4 outlined above leads us to our second result below dealing with CCE manifolds of general dimensions d.

As an application of Theorem 1.2, we are able to establish some global uniqueness result for the CCE metrics on X with prescribed conformal infinities constructed by Graham–Lee [14].

As a direct consequence, we have the following non-existence of CCE fill-ins result.

The paper is organized as follows: In Section 2, we recall some basic ingredients which will be used later in the proofs of main theorems and list some of their key properties, including in particular the estimates of the injectivity radius. In Section 3, we prove the boundary regularity for X when d is even. In Section 4, we present a different proof for the boundary regularity for all d dimensional CCE manifolds X which works for all d. In Section 5, we establish various compactness results for the adapted metrics and prove Theorems 1.1 and 1.2. In Section 6, we prove Theorem 1.3 of the uniqueness of Graham–Lee metrics. In Corollary 6.1 we establish as an application some gap phenomenon for classes of conformal invariants.

We remark that in the paper, we have provided separate arguments to gain the regularity of the compactified metrics on X in Section 3 (when d is even) and in Section 4 (for all d). In the rest of the paper i.e. in Sections 1, 2, 5 and 6, the arguments work for both even or odd d.

2.1 Basic properties of adapted metrics g*

Let v be a solution of (1.4). We define a class of adapted metrics g* by g * ≔ v 4 d − 4 g + when the dimension d is greater than 4. First, we recall some asymptotic properties of v.

This lemma is a special case of the general scattering theory on CCE manifolds as described in Graham–Zworski [25]. In below we will describe some properties of this adapted metric g*.

The result is a special case of a much more general result in Ref. [18]. To avoid introducing more notations, here we will present a self-contained proof.

We now recall the formula of the Ricci curvature under conformal change of metrics, applying to g* = ρ ² g ⁺, we get

Thus

Applying (1.4), we get

which in turn gives

and

We now recall another important property of the adapted metrics g* established in an earlier work of Case and Chang [18], Lemma 4.2.

We will see in Section 5 that property (2.4) implies the convergence on compact subsets of a sequences of rescaled adapated metrics, which is one of the key ingredients to establish the compactness results in Theorems 1.1 and 1.2.

2.2 Elliptic estimates for the adapted metrics

Let R _ikjl , W _ikjl , R _ij and R be Riemann, Weyl, Ricci, Scalar curvature tensors respectively. We recall on general Riemannian manifold (X, g) of dimension d, the fourth-order Bach tensor B is defined as

Recall also the Cotton tensor C is defined as

where A is the Schouten tensor. Recall also a relation between the divergence of Weyl tensor to the Cotton tensor, namely

Applying this relation (2.7), we can write the Bach tensor into the following form:

where Q ₁(Rm) is the quadratic term on Riemann curvature tensor

Thanks to the second Bianchi identity for the Weyl tensor (see for instance [16], [26]), we have

where Ψ ikjl ≔ C lkm,m g j i + C lmi,m g j k + C lik,j − C jkm,m g l i − C jmi,m g l k − C jik,l . A direct computation leads to rewrite the Bach Equation (2.8) in turns of the Weyl tensor as follows:

where K is a quadratic of curvatures and L _ikjl ≔ − B _ji g _kl − B _lk g _ij + B _li g _jk + B _kj g _il is some linear term on the Bach tensors.

We also recall that the adapted metric g* which we haven chosen in Section 2.1 is Q-flat, i.e. Q[g*] = 0, which can be expressed in the following form Refs. [27], [28]:

In the proof of Theorem 2.12 and Lemma 3.1 in Section 3, we will first estimate the Bach tensor, then incorporate the Q-flat property of g* into the Bach Equation (2.8) to derive estimates of the Ricci curvature of g* and into the Equation (2.9) to derive estimates of its Weyl curvature.

In order to estimate the Bach tensor and the Cotton tensor of g* in the interior of X, as g* is conformal to the Einstein metric g ⁺, we can simplify their expressions as (2.14) and (2.15) below.

(2.11) and (2.12) are derived by a routine computation.

If we apply Lemma 2.4 to the adapted metrics g* = ρ ² g ⁺, using the fact that both the Bach tensor and the Cotton tensor for the Einstein metric g ⁺ vanish, and the fact that W _jkil [g*] = ρ ² W _jkil [g ⁺], we obtain the following formulas for the Bach tensor and Cotton tensor for g*.

In the next two lemmas, we will derive some preliminary estimates of the curvatures of g* and prepare ourselves for the proof of the main results in Section 3.

We now recall some basic facts relating the behavior of the curvatures of g* on the boundary to that of the curvature of its boundary metric, which we denote by g ̂ .

We denote ∂ ₁ the outward unit boundary normal direction; α, β ∈ {2, …, d} the tangential directions on M = ∂X.

All the identities in Lemma 2.6 above are straightforward consequence of the Gauss-Codazzi equation and the fact that for the boundary of the adapted metric g* is totally geodesic. Similar results as in the statement has been established before when d = 4 in the earlier work of Ref. [16], Lemma 2.7. The proof for general dimensions is tedious but similar, which we will place in Appendix A.

The next lemma is the iteration process to express the higher order derivatives of the Ricci and Weyl curvatures of g* on the boundary in term of the curvature of the boundary metric; these formulas will be used in the proof of Lemma 3.1 in Section 3.

2.3 Some results in Riemannian geometry for CCE manifolds

One fundamental tool to achieve compactness results in Riemannian geometry is the Cheeger-Gromov convergence theory (see, for example Refs. [29], [30], for manifolds without boundary, and Refs. [31]–[35], for manifolds with boundary). For our purpose, here we recall some basic facts of the Cheeger-Gromov compactness theorem for manifolds with boundary.

Another important tool in Riemannian geometry is to find criteria to establish the no collapsing phenomenon. In the setting of conformal compact Einstein manifolds, we will achieve this by applying an inequality (2.15) recently discovered by Li-Qing-Shi ([13], see also Ref. [12]. This inequality plays an important role in our proof of Theorem 1.3 and Corollary 6.1.)

The last topic in this subsection concerns the injectivity radius estimates for manifolds with boundary. For our purpose we may always assume that the geometry of the boundary is compact in the Cheeger-Gromov sense.

The same proof in our earlier work [15], Lemma 3.1 can be modified to establish this lemma, we will skip the proof here.

We get a similar estimate like Lemma 2.10 for the lower bound of the injectivity radius.

The proof of the above lemma is also similar to the one of Ref. [15], Lemma 3.3. Here we omit the details.

2.4 Interior regularity in all dimensions

The interior regularity estimates for CCE manifolds is relatively well known, we will provide the result here just for the sake of completeness. First we recall the definition of harmonic radius on a Riemannian manifold with boundary see Ref. [34]:

Assume (X, g) is a complete Riemnnian d-dimensional manifold with the boundary ∂X. A local coordinates

is said to be harmonic if,

1. △x _i = 0 for all 1 ≤ i ≤ d in B(p, r) ⊂ X, when p ∈ X is in the interior;

2. Δx _i = 0 for all 1 ≤ i ≤ d in B(p, r) ∩ X and, on the boundary B(p, r) ∩ ∂X, (x ₂, x ₃, …, x _d ) is a harmonic coordinate in ∂X at p while x ₁ = 0, when p ∈ ∂X is on the boundary.

For α ∈ (0, 1) and M ∈ (1, 2), we define the harmonic radius r ^1,α (M) to be the biggest number r satisfying the following properties:

1. If dist(p, ∂X) > r, there is a harmonic coordinate chart on B(p, r) such that

   (2.18) M − 2 δ j k ≤ g j k ( x ) ≤ M 2 δ j k ,

   and

   (2.19) r 1 + α sup | x − y | − α | ∂ g j k ( x ) − ∂ g j k ( y ) | ≤ M − 1

   in B ( p , r 2 ) ̄ .

2. If p ∈ ∂X, there is a boundary harmonic coordinate chart on B(p, 4r) such that (2.18) and (2.19) hold in B ( p , 2 r ) ̄ .

4.1 Gauged Einstein equation

In Ref. [22], the authors use gauged Einstein equation to study the regularity issue of g ⁺ up to a diffeomorphism. Later on Biquard–Herzlich have established [38] a local version of the result. We now briefly describe the set-up of their method, then indicate the modifications to apply their method to our setting.

Let Z _R (p) denote a domain defined by (B.1) in Appendix B. We consider the nonlinear functional introduced by Biquard [21] defined on the d-dimensional open set Z _R (p) with p ∈ ∂X for two asymptotically hyperbolic metrics g ⁺ and k ⁺.

where B k + is a linear differential operator on symmetric (0,2) tensor, which is the infinitesimal version of the harmonicity condition

Here, δ denotes the divergence operator of 2-tensors, δ* the symmetrized covariant derivative of the vector field and d the exterior derivative.

We now recall the Lichnerowicz Laplacian △ _L on symmetric 2-tensors given by.

where

and

We have for any asymptotically hyperbolic metrics k ⁺

where D ₁ denotes the differentiation of F with respective to its first variable.

Recall k ⁺ is an asymptotically hyperbolic (AH) metric on X if k ⁺ is a conformally compact metric on X such that for some compactified metric k = φ ² k ⁺ there holds ‖∇φ‖ ≡ 1 on ∂X.

It is clear that for any asymptotically hyperbolic Einstein metrics g ⁺,

Suppose (X ^d , ∂X, g ⁺) is a conformally compact Einstein manifold of dimension d ≥ 4 with a conformal infinity (∂X, [h]) of the positive Yamabe type. Assume that our adapted metrics g* is in the C ³ and that we have

1. ‖ Rm g * ‖ C 0 ≤ 1 ;

2. there exists some r ₀ > 0 such that the injectivity radius i _int(X) ≥ r ₀, i _∂(X) ≥ 2r ₀, i(∂X) ≥ r ₀;

3. ‖ h ‖ C 6 ≤ N for some positive constants N > 0.

Hence, we can identify { p ∈ X ̄ , ρ ( p ) ≤ r 1 } = [ 0 , r 1 ] × ∂ X ⊂ d g * ( p , ∂ X ) ≤ r 0 for some r ₁ > 0 (we could decrease r ₁ if necessary) as a submanifold with boundary. We consider a C ⁴ AH metric on [0, r ₁/2] × ∂X and its compactification:

where h ⁽²⁾ is the Fefferman-Graham expansion term and intrinsically determined by the boundary metric h. Given 2R < r ₁/2, we look for a local diffeomorphism Φ: Z _R (p) → Z _2R (p) such that Φ*g ⁺ solves the gauged Einstein equation in Z _R/2(p)

We divide the boundary ∂Z _R (p)≔∂^∞ Z _R (p) ∪ ∂^int Z _R (p) = ({ρ = 0}∩ ∂Z _R (p)) ∪ ({ρ > 0}∩ ∂Z _R (p)). Given a CCE g ⁺ and a regular AH t ⁺ with the same conformal infinity on the local boundary Ψ p , R Y 1 ∞ = ∂ ∞ Z R ( p ) , we try to find a local diffeomorphism Φ: Z _R (p) → Z _2R (p) such that the gauged condition is satisfied in Z _R/2(p) up to the diffeomorphism Φ fixing the boundary ∂^∞ Z _R (p), that is

Here Y ₁ is defined by (B.2) in Appendix B and Y 1 ∞ ≔ Y ̄ 1 ∩ { ( 0 , x ′ ) } . Thus, the gauged Einstein Equation (4.2) is satisfied in Z _R/2(p). We know ρ ∈ C loc 3 , γ for all γ ∈ (0, 1) under the adapted harmonic coordinates for the metric g*. More precisely, we have the following result.

We could identify the neighborhood {p ∈ X|ρ(p) ≤ r ₁/2} of ∂X in X as [0, r ₁/2] × ∂X. In fact, let (θ ², …, θ ^d ) be the harmonic chart of ∂X. We extend them as harmonic functions (x ², …, x ^d ) in X so that a local chart of {p ∈ X|ρ(p) ≤ r ₁/2} could be given by (ρ, x ², …, x ^d ). In view of Lemma 4.1, such chart is C ^3,γ compatible with the harmonic coordinates of X. Thus, recall the C ⁴ compacitified AH manifold on [0, r ₁/2] × ∂X