Sun-Yung A Chang
email: chang@math.princeton.edu

AND Yuxin Ge

Jie Qing
email: qing@ucsc.edu

COMPACTNESS OF CONFORMALLY COMPACT EINSTEIN 4-MANIFOLDS II

.

1. Introduction 1.1. Statement of Improved results. Let X 4 be a smooth 4-manifold 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 of the boundary ∂X in X. A defining function r is a smooth nonnegative function on the closure X such that ∂X = {r = 0} and dr = 0 on ∂X. A conformally compact metric g + on X is said to be conformally compact Einstein (CCE) if, in addition,

Ric[g + ] = -ng + ,
where Ric[•] denotes the Ricci tensor of the Riemannian metric (sometimes we use also indice to indicate the dependence on the metric). The most significant feature of CCE manifolds (X, g + ) is that the metric g + is "canonically" associated with the conformal structure [ĝ] on the boundary at infinity ∂X, where ĝ = g| T ∂X . (∂X, [ĝ]) is called the conformal infinity of a conformally compact manifold (X, g + ). It is of great interest in both mathematics and theoretic physics to understand the correspondences between conformally compact Einstein manifolds (X, g + ) and their conformal infinities (∂X, [ĝ]), especially due to the AdS/CFT correspondence in theoretic physics (cf. Maldacena [START_REF] Maldacena | The large N limit of superconformal field theories and supergravity[END_REF], [START_REF] Maldacena | TASI 2003 Lectures on AdS/CFT[END_REF] and Witten [START_REF] Witten | Anti de Sitter space and holography[END_REF]).

The project we work on is to address the compactness issue of given a sequence of CCE 4-manifolds (X 4 , M 3 , {g + i }) with M = ∂X and {g i } = {r 2 i g + i } a sequence of compactified metrics, denote ĝi = g i | M , assume {ĝ i } forms a compact family of metrics in M , when is it true that some representatives ḡi ∈ [g i ] with {ḡ i | M } = {ĝ i } also forms a compact family of metrics in X? We remark that, for a CCE manifold, given any conformal infinity, a special defining function which we call geodesic defining function r exists so that |∇[ḡ]r| ≡ 1 in an asymptotic neighbor M × [0, ) of M . We also remark that the eventual goal to study the compactness problem is to show existence of conformal filling in for some classes of Riemannian manifolds as conformal infinity.

One of the difficulty to address the compactness problem is due to the existence of some "non-local" term. To see this, we look at the asymptotic behavior of the compactified metric g of CCE 4-manifold (X 4 , M 3 , g + ) with conformal infinity (M 3 , ĝ) ( [START_REF] Graham | Volume and Area renormalizations for conformally compact Einstein metrics, The Proceedings of the 19th Winter School "Geometry and Physics[END_REF], [START_REF] Fefferman | The ambient metric[END_REF]) which takes the form g := r 2 g + = h + g (2) r 2 + g (3) r 3 + g (4) r 4 + • • •• on an asymptotic neighborhood of M × (0, ), where r denotes the geodesic defining function of g. It turns out g (2) = -1 2 A[ĝ], where A[ĝ] := Ric[ĝ] -1 6 R[ĝ]ĝ denotes the Schouten tensor, Ric[ĝ] the Ricci tensor and R[ĝ] the scalar curvature respectively for the metric ĝ. Thus g (2) is determined by ĝ (we call such terms local terms), T r ĝg (3) = 0, while

g (3) α,β = - 1 3 ∂ n (Ric[g]) α,β
where α, β denote the tangential coordinate on M and ∂ n is the normal derivative with respect to the outward unit normal n of the boundary under the metric g, is a non-local term which is not determined by the boundary metric ĝ. We remark that ĝ together with g (3) determine the asymptotic behavior of g ( [START_REF] Fefferman | The ambient metric[END_REF], [START_REF] Biquard | Continuation unique à partir de l'infini conforme pour les métriques d'Einstein[END_REF]).

In an earlier work of Chang-Ge [16, Lemma 2.1], for a CCE manifold (X 4 , M 3 , g + ), we introduce the notion of 2-tensor S which on a 3-manifold M 3

(S[g]) α,β := ∇ i (W [g]) iαnβ + ∇ i (W [g]) iβnα -∇ n (W [g]) nαnβ - 4 3 H[g](W [g]) αnβ n
where W [g] denotes the Weyl tensor, H[g] the mean curvature on the boundary M , letter i is full indice and Greek indices α, β represent the tangential indices. When the compactified metric g has totally geodesic boundary, it takes the form:

(S[g]) α,β = 1 2 ∂ n Ric[g] α,β - 1 12 ∂ n R[g] ĝα,β .
The 2-tensor S is conformally invariant in the sense that

S[r 2 g] = r -1 S[g].
The connection of the S tensor to that of g (3) is that (see [16, Remark 2.2, (2.7)]): Under any compactification by a geodesic defining function r, g = r 2 g + has ∂ n R[g] = 0 on M , thus

(1.1) (S[g]) α,β = - 3 2 g (3)
α,β .

This shows that g (3) is also a local conformal invariant, which has been stated by Graham [START_REF] Graham | Volume and Area renormalizations for conformally compact Einstein metrics, The Proceedings of the 19th Winter School "Geometry and Physics[END_REF].

In [START_REF] Chang | Compactness of conformally compact Einstein manifolds in dimension 4[END_REF], we have also considered a special choice of compactification g * , which we named the Fefferman-Graham's (FG) compactification, defined by solving the PDE:

(1.2) -∆ g + w = 3 on X 4 .

17 Jul 2020 01:43:24 PDT 190527-YuxinGe Version 2 -Submitted to Adv. Math.

g * := e 2w g + with g * | M = g Y , the Yamabe metric on the conformal infinity of (X 4 , g + ).

We now state the first result of this paper.

Theorem 1.1. Suppose that X is a smooth oriented 4-manifold with boundary ∂X = S 3 . Let {g + i } be a set of conformally compact Einstein metrics on X. Assume the following conditions:

(1) The set {ĝ i } of Yamabe metrics that represent the conformal infinities lies in a given set C of metrics that is of positive Yamabe type and compact in C k,α Cheeger-Gromov topology with k ≥ 3 and with some α ∈ (0, 1).

(2) The FG compactifications {g * i = ρ 2 i g + i } associated with the Yamabe representatives {ĝ i } on the boundary satisfies:

lim r→0 sup i sup x∈∂X B(x,r) |S i |[g * i ]dvol[ĝ i ] = 0
(3) H 2 (X, Z) = 0. Then, the set {g * i } of FG compactifications (after diffeomorphisms that fix the boundary) forms a compact family in the C k,α Cheeger-Gromov topology.

We now explain the connection of the S tensor to other scalar curvature invariants for the metric g * , which plays a key role in the results in [START_REF] Chang | Compactness of conformally compact Einstein manifolds in dimension 4[END_REF] and in this paper. In the following, if there is no confusion, we drop the argument g for the various curvature tensors Ric, R, W , etc. On a 4-manifold (X 4 , g), a 4-th order curvature called the Q-curvature is defined as:

(1.3) Q[g] := - 1 6 R - 1 2 |Ric| 2 + 1 6 R 2 .
Q curvature is naturally associated with a 4th-order differential operator:

P [g] := ( ) 2 -div [( 2 3 Rg -2Ric)∇]
called Paneitz operator [START_REF] Paneitz | A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds[END_REF][START_REF] Branson | Estimates and extremal problems for the log-determinant on 4-manifolds[END_REF]. We remark that Paneitz operator is a special case of the family of GJMS ( [START_REF] Graham | Conformally invariant powers of the Laplacian. I. Existence[END_REF]) operators. The relation of the pair {Q, P } in 4 dimensions is like that of the well known pair {K, -∆} in 2 dimensions, where K denotes the Gaussian curvature:

-∆[g] + K[g] = K[e 2w g]e 2w on X 2 , P [g]w + Q[g] = Q[e 2w
g]e 4w on X 4 for conformal changes of the metric. For a 4-manifold (X 4 , g) with boundary, in the earlier works of Chang-Qing [START_REF] Chang | The Zeta functional determinants on manifolds with boundary I-the formula[END_REF][START_REF] Chang | The Zeta functional determinants on manifolds with boundary II-Extremum metrics and compactness of isospectral set[END_REF], in connection with the 4th order Q curvature, a 3rd order "non-local" boundary curvature T was introduced on ∂X to study the boundary behavior of g. The relation between the pair (Q, T ) is a generalization of that of the Dirichlet-Neumann pair (-∆, ∂ n ). The expression of T curvature is in general complicated, but in the special case when g is totally geodesic, the expression T take the simple form:

(1.4) T [g] := 1 12 ∂ n R.
17 Jul 2020 01:43:24 PDT 190527-YuxinGe Version 2 -Submitted to Adv. Math.

We now state the second result of our paper.

Theorem 1.2. Suppose that X is a smooth oriented 4-manifold with boundary ∂X = S 3 . Let {g + i } be a set of conformally compact Einstein metrics on X. Assume the same conditions (1) and (3) as in Theorem 1.1 and

(2 ) For the associated Fefferman-Graham's compactifications {g * i = e 2w i g + i } with the Yamabe representatives {ĝ i } on the boundary,

lim inf r→0 inf i inf x∈∂X B(x,r) T [g * i ]dvol[ĝ i ] ≥ 0.
Then, the set {g * i } is compact in C k,α norm up to diffeomorphisms that fix the boundary, provided k ≥ 7.

In the earlier work of Chang-Ge [START_REF] Chang | Compactness of conformally compact Einstein manifolds in dimension 4[END_REF], both Theorem 1.1 and Theorem 1.2 were established under the extra conditions on the uniform bounds of the first and second Yamabe constants for manifolds with boundary. The conditions are:

(4) There exists some positive constant C 5 > 0 such that the first Yamabe constant for the compactified metric

g i := ρ 2 i g + i is bounded uniformly from below by C 2 i.e. Y (X, ∂X, [g i ]) := inf U ∈C 1 X (|∇U | 2 + 1 6 R[g i ]U 2 )dvol[g i ] X U 4 dvol[g i ] 1 2 ≥ C 5 .
(5) There exists some positive constant C 6 > 0 such that the second Yamabe constant for the metric g i is bounded uniformly from below by C 6 , i.e.

Y b (X, ∂X, [g i ]) := inf U ∈C 1 X (|∇U | 2 + 1 6 R[g i ]U 2 )dvol[g i ] ∂X U 3 dvol[ĝ i ] 2 3 ≥ C 6 ,
where ĝi = g i | T ∂X . We remark Condition (1) in the earlier work [START_REF] Chang | Compactness of conformally compact Einstein manifolds in dimension 4[END_REF]Theorem 1.1] is stated slightly weaker, that is, the Yamabe constant Y (∂X, [ĝ i ]) is assumed to be non-negative.

In the current paper we managed to drop both conditions (4) and ( 5) from the statements of Theorems 1.1 and 1.2, which is accomplished by applying another round of blow-up analysis to reduce the situation to the earlier version of the theorems in [START_REF] Chang | Compactness of conformally compact Einstein manifolds in dimension 4[END_REF]. We will present the proof in section 3 of this paper. Once the curvature of metric g * i is bounded, we will prove the diameter is uniformly bounded in section 4 by some new arguments. 17 Jul 2020 01:43:24 PDT 190527-YuxinGe Version 2 -Submitted to Adv. Math.

Statement of new results

. Due to the nature of the problem in the CCE setting, natural conditions to imply the compactness of the solutions should be conformally invariant conditions, condition (1) in the statements of Theorems 1.1 and 1.2 is conformally invariant, condition (3) is topologically invariant hence a conformal invariant, but unfortunately the condition (2) in Theorem 1.1 and condition (2') in Theorem 1.2 are not. It is in this direction we now have new results where compactness is reached under some conformally invariant conditions; as a consequence we also establish some "uniqueness" result of conformal filling in for a special class of CCE manifolds with given conformal infinity.

Theorem 1.3. Suppose that X is a smooth oriented 4-manifold with boundary ∂X = S 3 . Let {g + i } be a set of conformally compact Einstein metrics on X. Assume the same condition (1) in Theorem 1.1. Then there is

δ 0 > 0 such that if either (2 ) X 4 (|W | 2 dvol)[g + i ] < δ 0 , or (2 ) Y (∂X, [ĝ i ]) ≥ Y (S 3 , [g S ]) -δ 0 ,
then the set {g * i } of the FG compactifications (after diffeomorphisms that fix the boundary) is compact in C k,α Cheeger-Gromov topology for some k ≥ 3.

In fact, for > 0 and α ∈ (0, 1), there is δ > 0, if g + is a conformally compact Einstein metric on X 4 with the conformal infinity (∂X, [ĝ]) and g * is the FG compactification associated with the Yamabe representative that belongs the set C in (1) in Theorem 1.1, and if (2 ) (or (2 )) holds for some δ ≤ δ 0 , then there is a diffeomorphism

φ i : X4 → B4 and φ i = Id : ∂B 4 = S 3 → ∂X 4 = S 3 satisfying (1.5) φ * i g * i -g F G C k,α (
B4 ) < where g F G is the FG compactification of the hyperbolic metric associated with a round metric on S 3 .

We will now relate the condition (2 ) in Theorem 1.3 to some other natural geometric conformal invariant, namely the "renormalized volume" in the CCE setting. Although the renormalized volume can be defined on CCE manifolds (X n+1 , ∂X, g + ) for any dimension n, we will here mainly recall some basic facts on CCE manifolds (X 4 , ∂X, g + ) when n = 3.

The concept of "renormalized volume" in the CCE setting was introduced by Maldacena [START_REF] Maldacena | The large N limit of superconformal field theories and supergravity[END_REF] (see also the works of Witten [START_REF] Witten | Anti de Sitter space and holography[END_REF], Henningson-Skenderis [START_REF] Henningson | The holographic Weyl anomaly[END_REF] and Graham [START_REF] Graham | Volume and Area renormalizations for conformally compact Einstein metrics, The Proceedings of the 19th Winter School "Geometry and Physics[END_REF]). On CCE manifolds (X 4 , M 3 , g + ) with geodesic defining function r, For n odd,

Vol g + ({r > }) = c 0 -3 + c 2 -1 + V + o(1).
We call the zero order term V the renormalized volume. V is independent of

g + ∈ [g + ],
and hence are conformal invariants.

17 Jul 2020 01:43:24 PDT 190527-YuxinGe Version 2 -Submitted to Adv. Math.

We now recall Gauss-Bonnet-Chern formula on compact 4-manifolds (X 4 , ∂X, g) with totally geodesic boundary.

(1.6) 8π 2 χ(X) = 1 4 X |W [g]| 2 dvol[g] + 4 X σ 2 (A[g])dvol[g],
where

σ 2 (A[g]) := 1 4 ( 1 6 R[g] 2 -1 2 |Ric[g]| 2
) is the second elementary symmetric function of the Schouten tensor A[g] := 1 2 (Ric[g] -1 6 R[g]g). We also recall an earlier result: Proposition 1.4. (M. Anderson [START_REF] Anderson | L 2 curvature and volume renormalization of the AHE metrics on 4-manifolds[END_REF], Chang-Qing-Yang [START_REF] Chang | On the renormalized volumes for conformally compact Einstein manifolds, (Russian)[END_REF], [START_REF] Chang | On the renormalized volumes for conformally compact einstein manifolds[END_REF]) On conformal compact Einstein manifold (X 4 , M 3 , g + ), we have

V = 2 3 X 4 σ 2 (A[g])dvol[g]
for any compactified metric g with totally geodesic boundary. Thus

(1.7) 8π 2 χ(X 4 , M 3 ) = 1 4 |W [g]| 2 dvol[g] + 6V.
We briefly recall the proof of above Proposition in Chang-Qing-Yang [START_REF] Chang | On the renormalized volumes for conformally compact Einstein manifolds, (Russian)[END_REF], as this is the crucial point that leads us to adopt the Fefferman-Graham's compactification to study the compactness problem of CCE manifolds.

Sketch proof of Proposition 1.4.

Lemma 1.5. (Fefferman-Graham [START_REF] Fefferman | Q-curvature and Poncaré metrics[END_REF]) Suppose (X 4 , ∂X, g + ) is conformally compact Einstein with conformal infinity (∂X, [ĝ]), fix ĝ ∈ [ĝ] and r its corresponding geodesic defining function. Consider the solution w to (1.2), then w has the asymptotic behavior w = log r + A + Br 3 near ∂X, where A, B are functions even in r, A| ∂X = 0, then

V = ∂X B| ∂X .
Lemma 1.6. (Chang-Qing-Yang [START_REF] Chang | On the renormalized volumes for conformally compact Einstein manifolds, (Russian)[END_REF]) With the same notation as in Lemma 1.5, Consider the metric g * = g w = e 2w g + , then g * is totally geodesic on boundary with (1)

Q[g * ] ≡ 0, (2) B| ∂X = 1 36 ∂ n R[g * ] = 1 3 T [g * ].
Proof. Recall we have g + is Einstein with Ric g + = -3g + , thus

P g + = (-∆ g + ) • (-∆ g + -2)
and Q[g + ] = 6. Therefore

P g + w + Q[g + ] = 0 = e 2w Q[g * ].
Assertion (2) follows from a straight forward computation using the scalar curvature equation and the asymptotic behavior of w. 17 Jul 2020 01:43:24 PDT 190527-YuxinGe Version 2 -Submitted to Adv. Math.

Applying Lemmas 1.5 and 1.6, we get

6V = 6 ∂X B| ∂X dvol[ĝ] = 1 6 ∂X ∂ n R[g * ]dvol[ĝ] = X Q[g * ]dvol[g * ] + 2 ∂X T [g * ]dvol[ĝ] = 4 X σ 2 (A[g * ])dvol[g * ].
For any other compactified metric g with totally geodesic boundary, X σ 2 (A[g])dvol[g] is a conformal invariant, and V is a conformal invariant, thus the result holds once for g * , holds for any such g in the same conformal class, which establishes Proposition 1.4.

We also recall some well known fact that (cf. [START_REF] Chang | On the topology of conformally compact Einstein 4-manifolds, Noncompact Problems at the intersection of Geometry[END_REF][START_REF] Chang | On the renormalized volumes for conformally compact Einstein manifolds, (Russian)[END_REF]). for a conformally compact Einstein 4-manifold with the conformal infinity of positive Yamabe type,

(1.8) V (X 4 , g + ) ≤ V (H 4 , g H ) = 4π 2
3 where the equality holds if and only if (X 4 , g + ) is isometric to (H 4 , g H ).

We now restrict our attention to class of CCE manifolds (B 4 , S 3 , g + ), in this class, for the model case when g + = g H , formulas for the specific FG g * metric can be computed straight forwardly.

Lemma 1.7. On (B 4 , S 3 , g H ),

g * = e (1-|x| 2 ) |dx| 2 on B 4 Q[g * ] ≡ 0, T [g * ] ≡ 2 on S 3 (g * ) (3) ≡ 0 and B 4 σ 2 (A[g * ])dvol[g * ] = 2 π 2 .
On (B 4 , S 3 , g), for a compact metric g with totally geodesic boundary, Gauss-Bonnet-Chern formula takes the form:

8π 2 χ(B 4 , S 3 ) = 8π 2 = B 4 ( 1 4 |W [g]| 2 + 4σ 2 (A[g]))dvol[g],
Thus we reached the following corollary of Theorem 1.3: Corollary 1.8. Let {X = B 4 , M = ∂X = S 3 , g + } be a 4-dimensional oriented CCE on X with boundary ∂X. Assume the boundary Yamabe metric h = h Y in the conformal infinity of positive type and Y (S 3 , [h]) > c 1 for some fixed c 1 > 0 and h is compact in C k,α norm with k ≥ 3. Let g * be the corresponding FG compactification. Then the following properties are equivalent:

(1) There exists some small positive number ε > 0 such that

X σ 2 (A[g * ])dvol[g * ] ≥ 2π 2 -ε.
(2) There exists some small positive number ε > 0 such that

X |W [g + ]| 2 dvol[g + ] ≤ 4ε.
17 Jul 2020 01:43:24 PDT 190527-YuxinGe Version 2 -Submitted to Adv. Math.

(3) There exists some small positive number ε 1 > 0 such that

Y (S 3 , [g c ]) ≥ Y (S 3 , [h]) > Y (S 3 , [g c ]) -ε 1
where g c is the standard metric on S 3 . (4) There exists some small positive number ε 2 > 0 such that for all metrics g * with boundary metric h same volume as the standard metric g c on S 3 , we have

T [g * ] ≥ 2 -ε 2 .
(5) There exists some small positive number ε 3 > 0 such that

|(g * ) (3) | ≤ ε 3 .
Where all the ε i (i = 1,2,3) tends to zero when ε tends to zero and vice versa for each i.

As an application of Theorem 1.3, we are able to establish the global uniqueness for the conformally compact Einstein metrics on B 4 with prescribed conformal infinities that very close to the conformal round 3-sphere (cf. [START_REF] Graham | Einstein metrics with prescribed conformal infinity on the ball[END_REF][START_REF] Lee | Fredholm operators and Einstein metrics on conformally compact manifolds[END_REF][START_REF] Li | Cap phenomena and curvature estimates for conformally compact Einstein manifolds[END_REF]). Namely, Theorem 1.9. For a given conformal 3-sphere (S 3 , [ĝ]) that is sufficiently close to the round one in C 3,α topology with α ∈ (0, 1), there is exactly one conformally compact Einstein metric g + on B 4 whose conformal infinity is the prescribed conformal 3-sphere (S 3 , [ĝ]).

Remark 1.10. We remark

• First, in Theorems 1.1 and 1.2, if we assume more assumptions on the topology in (3) H 1 (X, Z) = H 2 (X, Z) = 0, we can drop the assumption on the boundary ∂X = S 3 (see [START_REF] Chang | Compactness of conformally compact Einstein manifolds in dimension 4[END_REF]). • Second, in Theorems 1.3, we do not need the boundary condition ∂X = S 3 for the compactness result in the first part; for the second part, when the boundary condition ∂X = S 3 holds and δ 0 is small enough, we have the topology of X is the ball B 4 . Therefore, Theorem 1.9 holds even without the assumption X = B 4 . • Third, in Theorems 1.1 , 1.2 and 1.3, we have the compactness results in C 2,β topology for any β ∈ (0, 1) if we assume the compactness (or more generally boundedness) of the boundary metric ĝ in C 3 topology (or even in C 2,β topology). Hence, we could expect the uniqueness result in Theorem 1.9 holds in the C 3 topology (or even in C 2,β topology) (thanks to [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]Theorem 8.29]).

The rest of this paper is organized as follows: In Section 2, we recall some basic ingredients in the proofs and list some of their key properties. In Section 3, we prove the injectivity radius estimates which is the major technical step in the blow-up analysis in Riemannian geometry. In Section 4, we establish various compactness results for Fefferman-Graham's compactifications and prove Theorems 1.1, 1.2 and 1.3. Finally, in Section 5, we prove Theorem 1.9 to obtain the global uniqueness for the conformally compact Einstein metrics on B 4 constructed earlier in [START_REF] Graham | Einstein metrics with prescribed conformal infinity on the ball[END_REF][START_REF] Lee | Fredholm operators and Einstein metrics on conformally compact manifolds[END_REF].

Acknowledge. The authors thank Jason Lotay, Joel Fine and Felix Schulze for their comments of the first version of the draft. In particular, they point out a mistake in Lemma 3.1 in that version. 17 Jul 2020 01:43:24 PDT 190527-YuxinGe Version 2 -Submitted to Adv. Math.

Preliminaries

2.1. Fefferman-Graham's compactifications. Suppose that X is a smooth 4-manifold with boundary ∂X and g + is a conformally compact Einstein metric on X. Let g * = ρ 2 g + be the Fefferman-Graham's compactification, that is, w := log ρ satisfies the equation (1.2). The function ρ was first introduced in [START_REF] Fefferman | Q-curvature and Poncaré metrics[END_REF] then in [START_REF] Chang | On the renormalized volumes for conformally compact Einstein manifolds, (Russian)[END_REF] to study the renormalized volume. We recall some basic calculations for curvatures under conformal changes. Write g + = r -2 g for some defining function r and calculate

Ric[g + ] = Ric[g] + 2r -1 ∇ 2 r + (r -1 r -3r -2 |∇r| 2 )g. Then one has R[g + ] = r 2 (R[g] + 6r -1 r -12r -2 |∇r| 2 ).
Here the covariant derivatives is calculated with respect to the metric g (or Fefferman-Graham's compactification g * in the following). Therefore, for a Fefferman-Graham's compactification g * of a conformally compact Einstein metric g + , one has

(2.1) R[g * ] = 6ρ -2 (1 -|∇ρ| 2 ),
which in turn gives

(2.2) Ric[g * ] = -2ρ -1 ∇ 2 ρ and (2.3) R[g * ] = -2ρ -1 ρ.

Now we recall

Lemma 2.1. ( [START_REF] Case | On fractional GJMS operators[END_REF], [16, Lemma 4.2]) Suppose that X is a smooth 4-manifold with boundary ∂X and g + is a conformally compact Einstein metric on X with the conformal infinity (∂X, [ĝ]) of nonnegative Yamabe type. Let g * = ρ 2 g + be the Fefferman-Graham's compactification associated with the Yamabe metric ĝ of conformal infinity. Then the scalar curvature R[g * ] is positive in X. In particular,

(2.4) ∇ρ [g * ] ≤ 1.
2.2. Elliptic estimates for Bach-flat and Q-flat metrics. Next we recall from [START_REF] Chang | Compactness of conformally compact Einstein manifolds in dimension 4[END_REF] the ε-estimates for Fefferman-Graham's compactifications g * of conformally compact Einstein metrics g + . We will continue to use the 2-tensor S when deriving estimates for Fefferman-Graham's compactifications, which are Bach-flat and Q-flat metrics. We first recall Bach equations in 4 dimensions for Bach-flat metrics:

(2.5) ∆A ij - 1 6 R ,ij + R ikjl A kl -R ik A k j + 1 2 W ikjl A kl = 0,
where

A ij = 1 2 (R ij - 1 6 R g ij )
is the Schouten tensor, R ikjl and W ikjl are Riemann and Weyl curvature tensors respectively, and Q-flat equation: (2.7)

(2.6) ∆R = 3| Ric | 2 -R 2
∆W ijkl + ∇ l C kji + ∇ k C lij + ∇ i C jkl + ∇ j C ilk := K ijkl ,
where C ijk = A ij,k -A ik,j is the Cotton tensor and K = W * Rm+g * * W * A is a quadratic of curvatures (cf. [16, (2.5)]). Finally, to get estimates for the full Riemann curvature tensor Rm, one recalls that Rm = W + g * A.

The most important analytic tools for elliptic estimates here are Sobolev inequalities. The conditions ( 4) and ( 5) in Section 1 (cf. [16, Theorem 1.1]) essentially provide the following Sobolev inequality and trace Sobolev inequality respectively:

(2.8) ( X |u| 6 dvol[g * ]) 1 3 ≤ C s X (|∇u| 2 )dvol[g * ] and
(2.9)

( ∂X |u| 4 dvol[ĝ]) 1 2 ≤ C b X (|∇u| 2 )dvol[g * ]
for all u ∈ C 1 0 (B(p, r 0 )), where p is any point in X and r 0 > 0 is fixed. Moreover, a global trace Sobolev inequality holds (2.10)

( ∂X |u| 4 dvol[ĝ]) 1 2 ≤ C b X (|∇u| 2 + |u| 2 )dvol[g * ]
for all u ∈ C 1 0 (X). Lemma 2.2. ([16, Theorem 3.4]) Suppose that X is a smooth 4-manifold with boundary ∂X and g + is a conformally compact Einstein metric on X with the conformal infinity of positive Yamabe type. Let g * = ρ 2 g + be the Fefferman-Graham's compactification associated with the Yamabe metric of the conformal infinity. Assume the Sobolev inequality (2.8) holds for the Fefferman-Graham's compactification g * . Then there exists constants ε > 0 and where t γ is the first intersection of γ with the boundary ∂X. Then the interior injectivity radius is defined by i int (X, g) = inf{i int (p, g) : p ∈ X}. For p ∈ ∂X, let i ∂ (p, g) be the supremum of r such that the normal geodesic γ from p in the inward unit normal direction ν p is minimizing for any t ∈ [0, r]. Then the boundary injectivity radius is defined by

C k > 0 such that if Rm L 2 (B(p,r)) ≤ ε for a geodesic ball B(p, r) ⊂ X, then, for each k = 0, 1, 2, • • • , (2.11) sup B(p,r/2) |∇ k Rm| ≤ C k r k+2 B(p,r) |Rm| 2 dvol[g * ] 1
i ∂ (X, g) = inf{i ∂ (p, g) : p ∈ ∂X}.
The other equivalent definition for the boundary injectivity radius is that i ∂ (X, g) is the supremum of the height h of the Fermi coordinates from the boundary ∂X in X:

exp p (sν p ) : ∂X × [0, h) → X for p ∈ ∂X and s ∈ [0, h) (cf. [35] [22, Section3.6]).
Lemma 2.4. Let (X 4 , g) be a complete Riemannian 4-manifolds with totally geodesic boundary. Suppose that | Rm | ≤ k and that i int (X, g) ≥ i 0 , i ∂ (X, g) ≥ i 0 , and i(∂X) ≥ i 0 .

for a positive constant i 0 , where i(∂X) is the intrinsic injectivity radius of the boundary. Then the Sobolev inequalities (2.8) and (2.9) (resp. (2.10)) hold for uniform constants C s and C b (resp. C b ).

Proof. We consider the doubling X = X ∪ ∂X X: the union of two copies of X along the boundary ∂X where the second X is the reflexion of X. It is easy to see that i( X, g) ≥ i 0 . Then, (2.8) (local and global) simply follows from [14, Theorem 1] (see also [START_REF] Aubin | Problémes isopérimétriques et espaces de Sobolev[END_REF] and other related results [33, Theorem 3.14 and Lemma 3.17]).

For the trace Sobolev inequality (2.9), one may first use [35, Theorem A] to find uniform Lipschitz boundary local coordinate system in which the trace Sobolev inequality (2.9) is valid with uniform constant C b at least for the local version.

To prove that (2.10) holds globally, we work with a partition of unity associated with a countable coordinate chart covering {B(x i , δ/2)}, where (x i ) be a sequence of points in X, such that Then there exists N = N (n, k, i 0 ), depending on n, k, v, such that each point of X has a neighborhood which intersects at most N of the balls B(x i , δ)'s. This comes from Gromov-Bishop volume comparison theorem. Meanwhile, if let K be the total number of B(x i , δ/2) that intersects with B(p, r 0 ) ∩ ∂X, then K depends only of r 0 and δ.

X = ∪ i B(x i , δ/2) and B(x i , δ/4) ∩ B(x j , δ/4) = ∅ if i = j.
Let ξ be some non-negative cut-off function such that ξ(t) = 1 on [0, δ/2] and ξ(t) = 0 on [3δ/4, +∞), and it satisfies |ξ | ≤ C/δ on [0, +∞). Let α i (x) = ξ(d(x, x i )) and

η i = α i / m α m . Let u ∈ C 1 (X). We can estimate ∂X |u| 4 dvol[g| ∂X ] 1 2 ≤ ( i ∂X |η i u| 4 dvol[g| ∂X ] 1 4
) 2

≤ ( i ( X |∇(η i u)| 2 dvol[g]) 1
2

) 2 ≤ CKN X (|∇u| 2 + u 2 )dvol[g])
Thus the proof is complete.

Remark 2.5. In the recent paper [START_REF] Chen | Escobar-Yamabe compactifications for Poincaré-Einstein manifold and rigidity theorems[END_REF], the authors have established the remarkable inequality that

Y b (X, ∂X, [g * ]) 2 ≥ 6Y (∂X, [ĝ])
for any conformally compact Einstein manifold (X, g + ) with its conformal infinity of positive Yamabe type. We remark that as a direct consequence one can drop the assumption (5) in the statements of the main theorems in [START_REF] Chang | Compactness of conformally compact Einstein manifolds in dimension 4[END_REF] . In other words, the global trace-Sobolev inequality (5) (therefore (2.10)) is always available for any conformally compact Einstein manifold (X, g + ) with its conformal infinity of positive Yamabe type. Thus the effort in the current paper is to drop the assumption (4), although the same procedure also works to drop assumption [START_REF] Anderson | L2 curvature and volume renormalization of the AHE metrics on 4-manifolds[END_REF] in the statements of both Theorem 1.1 and Theorem 1.2.

2.4. Cheeger-Gromov convergences for manifolds with boundary. Our approach to establish the compactness of conformally compact Einstein 4-manifolds is to prove by contradiction. We will analyze and eliminate the causes of possible non-compactness by the method of blow-up. This method has been essential and powerful in many compactness problems in geometric analysis, particularly in Riemannian geometry. The fundamental tool in the context of Riemannian geometry is the so-called Cheeger-Gromov convergences of Riemannian manifolds developed from Gromov-Hausdorff convergences (see, for example, [START_REF] Cheeger | Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds[END_REF][START_REF] Anderson | Convergence and rigidity of manifolds under Ricci curvature bounds[END_REF], for Cheeger-Gromov convergences of Riemannian manifolds without boundary). In this subsection, for later uses in our paper, we will present the Cheeger-Gromov convergences for manifolds with boundary. Good references in the subject are for examples in [START_REF] Perales | A survey on the convergence of manifolds with boundary[END_REF][START_REF] Kodani | Convergence theorem for Riemannian manifolds with boundary[END_REF][START_REF] Knox | A compactness theorem for riemannian manifolds with boundary and applications[END_REF][START_REF] Wong | An extension procedure for manifolds with boundary[END_REF][START_REF] Anderson | Boundary regularity for the Ricci equation, geometric convergence, and Gelfand's inverse boundary problem[END_REF].

Let us first recall the definition of harmonic radius for a Riemannian manifold with boundary (cf. [START_REF] Perales | A survey on the convergence of manifolds with boundary[END_REF]). Assume (X, g) is a complete Riemnnian 4-manifold with the boundary ∂X. A local coordinates is said to be harmonic if,

(x 0 , x 1 , x 2 , x 3 ) : B(p, r) → Ω ⊂ R 4
• x i = 0 for all 0 ≤ i ≤ 3 in B(p, r) ⊂ X, when p ∈ X is in the interior;

• ∆x i = 0 for all 0 ≤ i ≤ 3 in B(p, r) ∩ X and, on the boundary B(p, r) ∩ ∂X, (x 1 , x 2 , x 3 ) is a harmonic coordinate in ∂X at p while x 0 = 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:

• If dist(p, ∂X) > r, there is a harmonic coordinate chart on B(p, r) such that (2.13) The following is the extension of the C 1,α convergence theorem of Anderson [START_REF] Cheeger | Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds[END_REF][START_REF] Anderson | Convergence and rigidity of manifolds under Ricci curvature bounds[END_REF] to manifolds with boundary (cf. [START_REF] Kodani | Convergence theorem for Riemannian manifolds with boundary[END_REF][START_REF] Anderson | Boundary regularity for the Ricci equation, geometric convergence, and Gelfand's inverse boundary problem[END_REF]).

M -2 δ jk ≤ g jk (x) ≤ M 2 δ jk and (2.14) r 1+α sup |x -y| -α |∂g jk (x) -∂g jk (y)| ≤ M -1 in B(p, r 2 ). • If p ∈ ∂X,
Lemma 2.6. ([6, Theorem 3.1]) Suppose that M(R 0 , i 0 , h 0 , d 0 ) is the set of all compact Riemannian manifolds (X, g) with boundary such that

|Ric X | ≤ R 0 , |Ric ∂X | ≤ R 0 i int (X) ≥ i 0 , i ∂ (X) ≥ 2i 0 , i(∂X) ≥ i 0 , Diam(X) ≤ d 0 , H Lip(∂X) ≤ h 0 ,
where Ric X (resp. Ric ∂X ) is the Ricci curvature of X (resp. the boundary), i(∂X) is the injectivity radius of the boundary, and H is the mean curvature of the boundary. Then M(R 0 , i 0 , h 0 , d 0 ) is pre-compact in the C 1,α Cheeger-Gromov topology for any α ∈ (0, 1).

Remark 2.7. We remark • First, in [START_REF] Anderson | Boundary regularity for the Ricci equation, geometric convergence, and Gelfand's inverse boundary problem[END_REF], it is showed that the harmonic radius r 1,α (M ) is uniformly bounded from below in M(R 0 , i 0 , h 0 , d 0 ) (cf. [6, Theorem 3.2.1]). • Second, it is easy to see that, after having harmonic coordinate charts with the uniform size, one has the pre-compactness in C k+2,α Cheeger-Gromov topology if the Ricci curvatures are bounded in C k,α norm and the boundaries are all totally geodesic (even for k = 0), which is the convergence theorem that is useful to us later (see [START_REF] Chang | Compactness of conformally compact Einstein manifolds in dimension 4[END_REF]). • Third, one may have the pre-compactness in the Cheeger-Gromov topology with base points if dropping the assumption on the diameter Diam(X).

Injectivity radii: blow-up before blow-up

Our main results in this section concern the injectivity radius estimates for manifolds with boundary. For our purpose we may always assume that the geometry of the boundary is compact in Cheeger-Gromov sense. The following is an easy consequence from [6, Theorem 3.1], which was mentioned as Lemma 2.6 in the previous Section 2.

Lemma 3.1. Suppose that (X 4 , g + ) is a conformally compact Einstein 4-manifold with the conformal infinity of Yamabe constant Y (∂X, [ĝ]) ≥ Y 0 > 0. And suppose that (X 4 , g * ) is the Fefferman-Graham's compactification associated with the Yamabe metric ĝ on the boundary such that the intrinsic injectivity radius i(∂X, ĝ) ≥ i o > 0, and that i ∂ (X, g * ) ≤ i int (X, g * ). Then there is a constant C ∂ > 0, depending of i 0 , such that

(3.1) max X |Rm|(i ∂ (X, g * )) 2 + i ∂ (X, g * ) ≥ C ∂
where Rm is Riemann curvature of g * .

Proof. We show by contradiction. Suppose otherwise there are a sequence of Fefferman-Graham's compactified Riemannian manifolds (X i , g * i ) such that max

X i |Rm[g * i ]|(i ∂ (X i , g * i )) 2 + i ∂ (X i , g * i ) → 0
and i(∂X i ) ≥ i 0 for some fixed positive number i 0 . We then rescale the metrics as follows:

ḡi = K -2 i g * i where K i = i ∂ (p i ) = i ∂ (X i , g * i )
for some p i ∈ ∂X i . Here we use the fact that the boundary injectivity radius i ∂ (•) is a continuous function on the boundary since the limit of minimizing geodesics is still minimizing geodesic. Recall the boundary of g * i is totally geodesic. On the other hand, because the curvature max

X i |Rm[ḡ i ]| = max X i |Rm[g * i ]|(i ∂ (X i , g * i )) 2 → 0,
by [START_REF] Kodani | Convergence theorem for Riemannian manifolds with boundary[END_REF]Lemma 6.3], there is a normal geodesic γ of length 2 such that γ is orthogonal to boundary ∂X i at γ(0) = p i and γ(2) = p i .

In light of Lemma 2.6, we may extract a subsequence (we will always use the same index for subsequences for convenience in this paper) (X i , ḡi , p i ) that converges to (X 4 ∞ , g ∞ , p ∞ ) in C 1,α Cheeger-Gromov topology. From the assumptions, it is easily seen that (X 4 ∞ , g ∞ , p ∞ ) is a complete flat metric manifold with the totally geodesic complete flat boundary (it is smooth in harmonic coordinates as demonstrated in [START_REF] Anderson | Convergence and rigidity of manifolds under Ricci curvature bounds[END_REF][START_REF] Anderson | Boundary regularity for the Ricci equation, geometric convergence, and Gelfand's inverse boundary problem[END_REF]). First, each connected component of the boundary (∂X ∞ , ĝ∞ ) is the Euclidean space R 3 because of i(∂X i ) ≥ i 0 . Second, by the Cheeger and Gromoll's splitting theorem, the complete metric g ∞ is a product metric on R 3 × (0, ∞) or R 3 × (0, 2). We claim the later case does not appear, that is, X ∞ = R 3 × (0, ∞). Let ρi be the Fefferman-Graham's defining function related to the metric ḡi . As in the proof of [START_REF] Chang | Compactness of conformally compact Einstein manifolds in dimension 4[END_REF]Lemma 4.3], we know, up to a subsequence, ρi → ρ ∞ and the hessian of ρ ∞ vanishes since g ∞ is flat. Moreover, ρ ∞ (x) ≥ Cdist g∞ (x, ∂X ∞ ) for some positive constant C > 0. In view of the boundary condition, ρ ∞ vanishes on R 3 ×{0}. Hence, ρ ∞ (x 0 , x 1 , x 2 , x 3 ) = αx 0 with some positive constant α > 0 since ∇ 2 ρ ∞ = 0. When ∂X ∞ has more than 2 connected components, this contradicts the fact ρ ∞ vanishes on ∂X ∞ . Therefore, we prove the claim. Now, on the other hand, there is a geodesic of length 2 in (X ∞ , g ∞ ) which are orthogonal to the boundary ∂X ∞ . This is a contradiction. Therefore, we have established the Lemma.

17 Jul 2020 01:43:24 PDT 190527-YuxinGe Version 2 -Submitted to Adv. Math.

Next we would like to get the lower bound estimates for the interior injectivity radius i int of a compact Riemannian manifold with boundary. The real reason for having no interior collapsing follows from the following recent work in [START_REF] Li | Cap phenomena and curvature estimates for conformally compact Einstein manifolds[END_REF].

vol g + (B(p, r)) vol g H 4 (B(r)) ≥ ( Y (∂X, [ĝ]) Y (S 3 , [g S ]) ) 3 2
As a consequence, we have Lemma 3.3. Suppose that (X 4 , g + ) is a conformally compact Einstein 4-manifold with the conformal infinity of Yamabe constant Y (∂X, [ĝ]) ≥ Y 0 > 0. And suppose that (X 4 , g * ) is the Fefferman-Graham's compactification associated with the Yamabe metric ĝ on the boundary such that the intrinsic injectivity radius i(∂X, ĝ) ≥ i o > 0, and that i ∂ (X, g * ) ≥ i int (X, g * ). Then there is a constant C int > 0, depending of Y 0 and i 0 , such that

(3.3) max X |Rm|(i int (X, g * )) 2 + i int (X, g * ) ≥ C int
where Rm is the Riemann curvature of g * .

Proof. Again, we will prove this lemma by contradiction. Assume otherwise, then there is a sequence of conformally compact Einstein 4-manifolds (X 4 i , g + i ) with the conformal infinity of Yamabe constants Y (∂X i , [ĝ i ]) ≥ Y 0 > 0, whose Fefferman-Graham's compactifications (X 4 i , g * i ) associated with the Yamabe metrics ĝi on the boundary are compact 4-manifolds with totally geodesic boundary and satisfy max

X i |Rm[g i ]|(i int (X i , g * i )) 2 + i int (X i , g * i ) → 0 and i(∂X i , ĝi ) ≥ i 0 .
Let us consider the rescaling ḡi

= K -2 i g * i for K i = i int (X i , g * i )
. Using [START_REF] Kodani | Convergence theorem for Riemannian manifolds with boundary[END_REF]Lemma 6.4] to the almost flat metrics ḡi , one sees that

1 = i(X i , ḡi ) = i int (p i , ḡi ) for some p i ∈ X i in the interior. Now, if K -1 i dist g * i (p i , ∂X i ) < ∞,
we are in the same situation as in the proof of Lemma 3.1 and derive the contradiction by [START_REF] Kodani | Convergence theorem for Riemannian manifolds with boundary[END_REF]Lemma 6.4] (there would be a closed geodesic of length 2 in the Euclidean half space).

Therefore we may assume that

K -1 i dist g * i (p i , ∂X i ) = dist ḡi (p i , ∂X i ) → ∞. Thus the limit space (X 4 ∞ , g ∞ , p ∞
) is a complete flat manifold with no boundary, but, with a simple closed geodesic of length 2. We claim that (X 4 ∞ , g ∞ , p ∞ ) is of Euclidean volume growth in dimensions 4. This would be a contradiction, since such flat manifold would be a product of a circle and a flat manifold of dimension 3, which would not be able to support the Euclidean volume growth in dimensions 4. 17 Jul 2020 01:43:24 PDT 190527-YuxinGe Version 2 -Submitted to Adv. Math.

To finish the proof of the claim that (X 4 ∞ , g ∞ , p ∞ ) is of Euclidean volume growth in dimensions 4, that is,

(3.4) vol[g ∞ ](B g∞ (p ∞ , r)) ≥ c v r 4
for some fixed c v and any r > 0. First let us prove the following claim.

Claim: ρi (p i ) → ∞, where ρi = K -1 i ρ i and K i = i int (X i , g * i ).
Proof. Assume otherwise that there is a constant ρ0 > 0 such that ρi (p i ) ≤ ρ0 for all i.

Due to (2.1) at the beginning of the Section 2.1, we have

1 -|∇[ḡ i ]ρ i | 2 = 1 6 R[ḡ i ]ρ 2 i ≥ 0,
where the covariant derivatives is calculated with respect to the background metric ḡi . Let us denote

i = max{|Rm[ḡ i ]|, (dist ḡi (p i , ∂X i )) -1 } → 0.
Then we obtain ρi (x) ≤ 1 2

-1 2 i + ρi (p i ) for all x ∈ B ḡi (p i , 1 2 -1 i ) ⊂ X i since |∇[ḡ i ]ρ i | ≤ 1. This in turn implies 1 ≥ |∇[ḡ i ]ρ i | ≥ 1 2 for all x ∈ B ḡi (p i , 1 2 -1
i ) ⊂ X i , at least for i sufficiently large. Therefore, along the integral curve γ(t) of the gradient ∇[ḡ i ]ρ i from p i , we may derive ρi (γ(t)) ≤ ρ0 -t 2 < 0 when t > 2ρ 0 , which is a contradiction since γ(t) ∈ X for any t ∈ (2ρ 0 , 1 2 -1 i ). So the proof of this claim is complete.

Now let

s i = min{dist ḡi (p i , ∂X i ), ρi (p i )} → ∞.
We find, for each x ∈ B ḡi (p i , s i 2 ),

|ρ i (x) -ρi (p i )| ≤ 1 2 ρi (p i ), which implies, 1 2 ≤ ρi (x) ρi (p i ) ≤ 3
i ](B ḡi (p i , r)) ≥ vol[ḡ i ](B g + i (p i , 2 3 (ρ i (p i )) -1 r)) ≥ 1 (2(ρ i (p i )) -1 ) 4 vol[g + i ](B g + i (p i , 2 3 (ρ i (p i )) -1 r)) ≥ C 1 (2(ρ i (p i )) -1 ) 4 vol[g H 4 ](B g H 4 ( 2 3 (ρ i (p i )) -1 r)) ≥ c v r 4
for a fixed constant c v that is independent of i. Passing to the limit as i → ∞, we get the desired inequality (3.4) on the limit space (X 4 ∞ , g ∞ , p ∞ ). So the proof is complete.

On compactness of Fefferman-Graham's compactifications

Based on the preparation in the previous sections we are ready to establish the compactness of Fefferman-Graham's compactifications on conformally compact Einstein 4manifolds which were stated in the introduction. The approach follows closely from the corresponding results in [START_REF] Chang | Compactness of conformally compact Einstein manifolds in dimension 4[END_REF]. The difference from [START_REF] Chang | Compactness of conformally compact Einstein manifolds in dimension 4[END_REF] is that, in light of the injectivity radius estimates in the previous section, Sobolev inequality and trace Sobolev inequality are all available for the rescaled metrics with bounded curvature, while Sobolev inequality and trace Sobolev inequality are parts of the assumptions in the main compactness theorem in [START_REF] Chang | Compactness of conformally compact Einstein manifolds in dimension 4[END_REF]. Readers are referred to [START_REF] Chang | Compactness of conformally compact Einstein manifolds in dimension 4[END_REF] for more details. To begin the proof, we will first establish some bounded curvature estimates. Lemma 4.1. Suppose that {(X 4 i , g + i )} is a sequence of conformally compact Einstein 4manifolds satisfying the assumptions (1), [START_REF] Anderson | Convergence and rigidity of manifolds under Ricci curvature bounds[END_REF], and (3) in Theorem 1.1. Then there is a positive constant K 0 such that, for the Fefferman-Graham's compactifications {(X 4 i , g * i )} associated with the Yamabe metric ĝi of the conformal infinity

(∂X i , [ĝ i ]) (4.1) max X i |Rm[g * i ]| ≤ K 0 for all i.
Proof. Suppose otherwise that there is a subsequence {(X 4 i , g + i )} satisfying

K i = max X i |Rm[g * i ]| → ∞. Let K i = K i (p i ) = |Rm[g * i ]
|(p i ) for some p i ∈ X i . Then we consider the rescaling (X 4 i , ḡi = K i g * i , p i ). We claim for the metrics ḡi the derivatives of curvature ∇Rm[g * i ] is uniformly bounded. For this purpose, we recall the curvature tensor satisfies some elliptic PDE with the

               R = R 2 -3|Ric| 2 in X R = 3 R on ∂X A -1 6 ∇ 2 R = Rm * A in X A αβ = Âαβ , A αn = 0, A αn , A nn = R 4 on ∂X W = Rm * W + g * W * A + L(Ric) in X W = 0 on ∂X,
where L is a linear differential operator of second order. In view of [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]Theorem 8.33] we have the boundedness of the curvature tensor in C 1,α topology since the curvature tensor on the boundary Rm is bounded in C 1,α topology.

No boundary Blow-up: Let us first consider the cases where

dist ḡi (p i , ∂X i ) < ∞.
For the pointed manifolds (X i , ḡi , p i ) with boundary, in the light of all the preparations in the previous sections, particularly Lemma 2.6 and Remark 2.7, we have Cheeger-Gromov convergence

(X 4 i , ḡi , p i ) → (X 4 ∞ , g ∞ , p ∞ ) in C k,α
Cheeger-Gromov topology (for a subsequence if necessary), where the limit space is a complete Bach-flat and Q-flat manifold with a totally geodesic boundary ∂X ∞ ; the boundary (∂X ∞ , ĝ∞ ) is simply the Euclidean space R 3 because i(∂X i ) ≥ i 0 > 0; and

|Rm[g ∞ ]|(p ∞ ) = 1.
To derive the a priori estimates for Cheeger-Gromov convergence, one applies theestimates in Lemma 2.2 and Lemma 2.3, where Sobolev inequality (2.8) and trace Sobolev inequality (2.9) are established in Lemma 2.4. The injectivity radii estimates that are needed for ḡi to satisfy Sobolev and trace Sobolev are given in Lemma 3.1 and Lemma 3.3. Now, clearly, to finish the proof is to show that the limit space (X 4 ∞ , g ∞ , p ∞ ) is the Euclidean half space. For the convenience of readers, we very briefly sketch the proof from [START_REF] Chang | Compactness of conformally compact Einstein manifolds in dimension 4[END_REF]. One first needs to show that ρi → ρ ∞ where ρ ∞ satisfies

• g + ∞ = ρ -2 ∞ g ∞ is a (partially) conformally compact Einstein metric on X 4 ∞ whose conformal infinity is the Euclidean space R 3 ; • -∆ g + ∞ log ρ ∞ = 3.
Then, by Condition (2) in Theorem 1.1, one shows that g + ∞ is locally hyperbolic space metric nearby the infinity ∂X 4 ∞ = R 3 based on the unique continuation therem in [START_REF] Biquard | Continuation unique à partir de l'infini conforme pour les métriques d'Einstein[END_REF][START_REF] Biquard | Analyse sur un demi-espace hyperbolique et poly-homogénéité locale[END_REF]. (at least for some subsequence). Notice that,

Finally one concludes that ρ ∞ = x 0 , since (X 4 ∞ , g + ∞ ) is hyperbolic space in half space model (R 4 + , |dx| 2 x 2 0 ) for R 4 + = {x = (x 0 , x 1 , x 2 , x 3 ) ∈ R 4 : x 0 > 0}, which implies that (X 4 ∞ , g ∞ )
K i = max X i |Rm[g * i ]| = |Rm[g * i ]|(p i )
for some p i ∈ X in the interior. Proceeding as the above boundary cases, one has the Cheeger-Gromov convergence

(X 4 i , ḡi , p i ) → (X 4 ∞ , g ∞ , p ∞ ) in C k,α
Cheeger-Gromov topology. The proof in these cases follows from [START_REF] Chang | Compactness of conformally compact Einstein manifolds in dimension 4[END_REF]. We again very briefly sketch the proof that is more or less from [START_REF] Chang | Compactness of conformally compact Einstein manifolds in dimension 4[END_REF]. One first derives from (2.1) that

R[ḡ i ] = ρ-2 i (1 -|∇[ḡ i ]ρ i | 2 ) and shows that • ρi (x) ≥ Cdist ḡi (x, ∂X i ). (cf.
Step 2 in the proof of [16, Lemma 4.9]). Then, consequently,

• R ∞ = 0, and • g ∞ is Ricci-flat from being Q-flat and scalar flat in the light of the Q-curvature equation (2.6). (cf.

Step 3 of the proof of [16, Lemma 4.9]). Thus, (X ∞ , g ∞ ) is a complete Ricci-flat 4-manifold with no boundary. At this point, as argued in [START_REF] Chang | Compactness of conformally compact Einstein manifolds in dimension 4[END_REF], first, due to the recent work in [START_REF] Cheeger | Regularity of Einstein manifolds and the codimension 4 conjecture[END_REF], one concludes that (X ∞ , g ∞ ) is a complete ALE Ricci flat 4-manifold. By the assumptions, the doubling of X is a homological sphere. By a topological result due to Crisp-Hillman ([26, Theorem 2.2]), (X ∞ , g ∞ ) at the infinity is asymptotic to S 3 /Γ with Γ = {1} or Γ = Q 8 (quaternion group) or Γ the perfect group (that is, S 3 /Γ is a homology 3-sphere). By the Chern-Gauss-Bonnet formula and the signature formula, we obtain the desired contradiction. For more details see [START_REF] Chang | Compactness of conformally compact Einstein manifolds in dimension 4[END_REF] section 4.3.

We now begin the proof of Theorem 1.1.

With the curvature bound (4.1), the injectivity radius estimates in Lemma 3.1 and Lemma 3.3, the -regularities Lemma 2.2 and Lemma 2.3, one last piece that is needed to apply the Cheeger-Gromov convergences for manifolds with boundary in Lemma 2.6 and Remark 2.7 to finish the proof of Theorem 1.1 is the following diameter bound. Lemma 4.2. Under the assumptions in Theorem 1.1, the diameters of the Fefferman-Graham's compactifications g * i are uniformly bounded. Proof. The proof of this lemma under the assumption that the first Yamabe invariant is uniformly bounded from below is obtained in [16, Section 5: The proof of Theorem 1.1]. However, we do not know if one has the suitable Euclidean Sobolev type inequality in actual setting. This makes the problem is more delicate. Here we give a different approach to overcome the difficulty.

We have already proved the family of metrics g * i has the bounded curvature in C 1 so that the arguments given in [START_REF] Chang | Compactness of conformally compact Einstein manifolds in dimension 4[END_REF]Section 4.4: The proof of Theorem 4.4] yields the bound in C k+1 norm. In view of Lemmas 3.1 and 3.3, the boundary radius and the interior one are uniformly bounded from below. Therefore, for all i, for all x ∈ X, we have 17 Jul 2020 01:43:24 PDT 190527-YuxinGe Version 2 -Submitted to Adv. Math. vol(B g * i (x, 1)) ≥ C > 0 for some constant C > 0 independent of i, x, that is, there is non-collapse. We prove the diameter is uniformly bounded from above by contradiction. Suppose that the diameter diam(g * i ) tends to the infinity. By Cheeger-Gromov-Hausdorff compactness theory, up to diffeomorphisms fixing the boundary, (X i , g * i ) converges to some complete non-compact manifold (X ∞ , g ∞ ) with the boundary. We divide the proof in 5 steps.

Step 1. There exists some C > 0 such that ρ i ≥ C provided d g * i (x, ∂X) ≥ 1 and

d g * i (x, ∂X) ≤ Cρ i (x) provided 0 ≤ d g * i (x, ∂X) ≤ 1.
Thus the limit metric is conformal to an asymptotic hyperbolic Einstein manifold. Moreover, there exists some constant

C 1 > 0 independent of i such that |Rm[g * i ]| 2 dvol[g * i ] ≤ C 1 .
The first part of the claim can be proved in the same way as in [16, Section 4: the proof of Lemma 4.4]. The second part is proved in [16, Section 5: the step 2 of the proof of Theorem 1.1]. Without loss of generality, assume the boundary injectivity radius is bigger than 1.

Step 2. There exists some constant C 2 > 0 independent of i such that

(4.3) {x,d g * i (x,∂X)≥1} ρ -2 i (x)dvol[g * i ] ≤ C 2 .
Thanks of (2.1) and (2.3), we infer

-log ρ i = R i 2 + |∇ρ i | 2 ρ 2 i = 3 (1 -|∇ρ i | 2 ) ρ 2 i + |∇ρ i | 2 ρ 2 i
Integrating on the set {x, d g * i (x, ∂X) ≥ 1}, we obtain

{x,d g * i (x,∂X)≥1} (3 (1 -|∇ρ i | 2 ) ρ 2 i + |∇ρ i | 2 ρ 2 i )dvol[g * i ] = {x,d g * i (x,∂X)=1} 1 ρ i ∇ρ i , ν
where ν is the outside normal vector on the boundary {x, d g * i (x, ∂X) = 1}. By Step 1, we know ρ i is uniformly bounded from below on the set {x, d g * i (x, ∂X) = 1}. Together the facts the curvature of g * i is bounded and the boundary (∂X i , ĝi ) is compact, we infer for some positive constant

C > 0 {x,d g * i (x,∂X)≥1} 3 (1 -|∇ρ i | 2 ) ρ 2 i )dvol[g * i ] ≤ C, and {x,d g * i (x,∂X)≥1} |∇ρ i | 2 ρ 2 i dvol[g * i ] ≤ C since |∇ρ i | ≤ 1.
Combining these estimates, the desired claim yields.

Step 3. We have lim

x→∞ ρ ∞ (x) = +∞ Letting i → ∞ in (4.3), we get (4.4) {x,dg ∞ (x,∂X)≥1} ρ -2 ∞ (x)dvol[g ∞ ] ≤ lim i {x,d g * i (x,∂X)≥1} ρ -2 i (x)dvol[g * i ] ≤ C 2 .
17 Jul 2020 01:43:24 PDT 190527-YuxinGe Version 2 -Submitted to Adv. Math.

For all ε > 0, there exists A > 0 such that

{x,dg ∞ (x,∂X)≥A} ρ -2 ∞ (x)dvol[g ∞ ] ≤ ε
Therefore, for any y with d g∞ (y, ∂X) ≥ A + 1, we can estimate

B g∞ (y,1) ρ -2 ∞ (x)dvol[g ∞ ] ≤ {x,dg ∞ (x,∂X)≥A} ρ -2 ∞ (x)dvol[g ∞ ] ≤ ε so that ( sup B g∞ (y,1) ρ ∞ ) -2 V ol(B g∞ (y, 1)) ≤ ε that is, sup B g∞ (y,1) ρ ∞ ≥ Cε -1/2
Together with Lemma 2.1, we deduce inf

B g∞ (y,1) ρ ∞ ≥ sup B g∞ (y,1) ρ ∞ -1 ≥ Cε -1/2 -1
Finally, we prove Step 3.

Step 4. We claim that there exists some c v > 0 such that for any p ∈ X ∞ and for any

r < 1 2 ρ ∞ (p) (4.5) V ol[g ∞ ](dist(B g∞ (p, r))) ≥ c v r 4
Let p i ∈ X i such that p i → p. First we remark that dist g * i (p i , ∂X i ) ≥ ρ i (p i ) because of Lemma 2.1. As in the proof of the end of Section 3, we have

V ol[g * i ](dist(B g * i (p i , r))) ≥ c v r 4 ,
where c v is some positive constant independent of i. Letting i → ∞, the claim is proved.

Step 5. A contradiction.

On choose p ∈ X ∞ such that ρ ∞ (p) is sufficiently large. We fix r = (ρ ∞ (p)) 2/3 . Using the results in Steps 2 and 4, we get ρ ∞ (p) (4.6) ( sup

B g∞ (p,r) ρ ∞ ) -2 V ol[g ∞ ](B g∞ (p, r)) ≤ B g∞ (p,r) ρ -2 ∞ (x)dvol[g ∞ ] ≤ C 2
so that for some positive contsant C > 0 there holds (4.7) sup

B g∞ (p,r) ρ ∞ ≥ Cr 2 = C(ρ ∞ (p)) 4/3
On the other hand, it follows from Lemma 2.1, we deduce (4.8) inf The proof of Theorem 1.2 is quite similar to the corresponding theorem in [START_REF] Chang | Compactness of conformally compact Einstein manifolds in dimension 4[END_REF] albeit the removing of conditions ( 4) and [START_REF] Anderson | L2 curvature and volume renormalization of the AHE metrics on 4-manifolds[END_REF]. We leave the details to the readers. The proof of Theorem 1.3 is different as we have no informations on the S-tensor or T curvature in the statement of the theorem. Here we will present the proof.

B g∞ (p,r) ρ ∞ ≥ sup B g∞ (p,r) ρ ∞ -r so that ρ ∞ (p) + (ρ ∞ (p)) 2/3 = ρ ∞ (p) + r ≥ C(ρ ∞ (p))
Proof of Theorem 1.3. Again we will prove by contradiction. Let {g + i } be a set of conformally compact Einstein metrics on X 4 and {g * i } corresponding Fefferman-Graham's compactifications associated with the Yamabe metrics ĝi ∈ C, where C is compact in C k,α Cheeger-Gromov topology as given in (1) in Theorem 1.1. Assume that

(2 ) either Y (∂X, [ĝ i ]) → Y (S 3 , [g S ]) or X (|W | 2 dvol)[g + i ] → 0, but (X, g * i )
does not converges in C k,α Cheeger-Gromov topology. If the interior blow-up were to happen, then, it is easily seen that it would be a contradiction with the fact that any possibly limit space is flat due to (2 ), in light of the rigidity in Gromov-Bishop's volume comparison principle or simply the limit metric is both Ricci flat and locally conformally flat. If the boundary blow-up were to happen, then it is again easily seen that it would be a contradiction with the fact that any possibly limit space would be with g + ∞ being hyperbolic. Therefore, by the proof of Theorem 1.1, one concludes that (X, g * i ) converges to the Fefferman-Graham's compactification of hyperbolic space in C k,α Cheeger-Gromov topology for some α ∈ (0, 1), from which we reached a contradiction.

Before ending this section, we turn to an important fact which is a consequence of the compactness result in Theorem 1.1, and in fact is an improved statement of the compactness for conformally compact Einstein metrics with the same conformal infinity. We will later use this fact (Theorem 4.4) to establish the uniqueness result in section 5. To state the result, we will use the notion of weighted spaces of tensors, which we refer the readers to [START_REF] Lee | Fredholm operators and Einstein metrics on conformally compact manifolds[END_REF] (see also [START_REF] Graham | Einstein metrics with prescribed conformal infinity on the ball[END_REF]). We first recall the following expansions in terms of the Fermi coordinate from the boundary, which we have stated in the introduction of this paper in terms of geodesic coordinates. The expansion in this form is motivated by an observation in [START_REF] Maldaceana | Einstein gravity from conformal gravity[END_REF]. Lemma 4.3. Let (X 4 , g) be a Bach-flat and Q-flat 4-manifold with the totally geodesic boundary. Then, in the Fermi coordinate from the boundary, one has g = dr 2 + g r and the expansion (4.9)

g r = ĝ + g (2) r 2 + g (3) r 3 + • • • ,
where g (2) is a curvature of ĝ = g| T ∂X and g (3) is not local.

Theorem 4.4. Suppose that, for two sequences of conformally compact Einstein metrics g + i and h + i that have the same conformal infinity of positive Yamabe type and satisfy the assumption in Theorem 1.1 (or the assumptions in Theorem 1.3). Then, for a weight δ ∈ (0, 3), there are subsequence (possibly after diffeomorphisms ψ i and φ i that fix the Suppose that g * i (resp. h * i ) (a subsequence) converges in C 3,α Cheeger-Gromov topology. Now, let us align the distance functions to be the same for all g * i (resp. h * i ) in this subsequence by diffeomorphisms ψ i (resp. φ i ) that fix the boundary, and get

ψ * i g * i -φ * i h * i = O(r 3
). for any i from Lemma 4.3. If necessary, extract a subsequence, for δ ∈ (0, 3) and any > 0, there is an index N , for i, j ≥ N ,

(ψ * i g * i -φ * i h * i ) -(ψ * j g * j -φ * j h * j ) C 2,α ( X∞) < 1 2 .
For any fixed δ < 3, one gets

ψ * i g * i -φ * i h * i C 2,α ≤ Cr 3 ≤
r δ over the region {r ≤ r } for some small r > 0 such that Cr 3-δ ≤ (C is independent of i due to the compactness in Theorem 1.1 ( resp. Theorem 1.3) and Lemma 4.3, and the sizes of Fermi coordinates for g * i has a uniform lower bound again follows from Theorem 1.1 (resp. Theorem 1.3)); while one gets

(ψ * i g * i -φ * i h * i ) -(ψ * j g * j -φ * j h * j ) C 2,α (
X∞) ≤ r δ over the rest {r ≥ r } by setting N larger, in the light of Theorem 1.1 (resp. Theorem 1.3). It is then easily seen that the corresponding ψ * i g + i -φ * i h + i converges in the weighted space C 2,α δ (X ∞ ) with δ ∈ (0, 3). This completes the proof of Theorem 4.4.

Uniqueness of Graham-Lee solutions in dimension 4

In this section we derive the global uniqueness result Theorem 1.9 based on some result in the recent work [START_REF] Li | Cap phenomena and curvature estimates for conformally compact Einstein manifolds[END_REF].

Proof of Theorem 1.9. We will prove this by contradiction. Assume otherwise there is a sequence of conformal 3-sphere (S 3 , [ĝ i ]) that converges to the round sphere such that, for each i, there exist two non-isometric conformally compact Einstein metrics g + i and h + i .

We first claim that, for a subsequence, both g + i and h + i converge to the hyperbolic space in C 3,α Cheeger-Gromov sense due to Theorem 1.2 and the uniqueness result when the conformal infinity is the standard sphere [START_REF] Qing | On the rigidity for conformally compact Einstein manifolds[END_REF][START_REF] Li | Cap phenomena and curvature estimates for conformally compact Einstein manifolds[END_REF]. 17 Jul 2020 01:43:24 PDT 190527-YuxinGe Version 2 -Submitted to Adv. Math.

Next, according to the proof of Theorem 4.4, we actually can conclude that, after some diffeomorphisim that fix the boundary φ i and ψ i ,

ψ * i g + i -φ * i h + i C 2,α
δ (H 4 ) → 0 for any δ ∈ (0, 3) and some subsequence. In other words, in this subsequence, the two distinct conformally compact Einstein metrics g + i and h + i are arbitrarily close to each other in weighted spaces, as long i is sufficiently large. We will show this is not possible by applying the local uniqueness result via the implicit function theorem on weighted norm space established in [START_REF] Graham | Einstein metrics with prescribed conformal infinity on the ball[END_REF][START_REF] Lee | Fredholm operators and Einstein metrics on conformally compact manifolds[END_REF]. We now give more details. We have for any asymptotically hyperbolic Einstein metrics h

+ D 1 F (h + , h + ) = 1 2 ( L + 6),
where D 1 denotes the differentiation of F with respective to its first variable, and where the Lichnerowicz Laplacian L on symmetric 2-tensors is given by

L := ∇ * ∇[g + ] + 2 • Ric[g + ] -2 • Rm[g + ];
where

• Ric[g + ](u) ij = 1 2 (R ik [g + ]u j k + R jk [g + ]u i k ),
and

• Rm[g + ](u) ij = R ikjl [g + ]u kl .
It is clear F (g + i , g + i ) = 0 We now divide the proof in 2 steps.

Step 1. Claim. We could find a diffeomorphism ϕ i of class C 3,α (equal to the identity on the boundary), such that F (ϕ * i h + i , g + i ) = 0 Moreover ϕ i (x) -x C 3,α → 0 and ϕ * i h + i -g + i C 2,α δ → 0 for some δ ∈ (2 + α, 3).

17 Jul 2020 01:43:24 PDT 190527-YuxinGe Version 2 -Submitted to Adv. Math.

It is sufficient to check this infinitesimally: the diffeomorphism group acts infinitesimally on g + i by taking the covector field X i to the symmetrized covariant derivative (δ g + i ) * X i , so the problem to solve is

B g + i ((δ g + i ) * X i ) = -B g + i (h + i ). Recall B g + i (δ g + i ) * = 1 2 ((∇) * ∇[g + i ] -Ric[g + i ]
), On the other hand, a direct calculation leads to (Proposition 2.5 [START_REF] Graham | Einstein metrics with prescribed conformal infinity on the ball[END_REF])

B g + (h + ) = ρ -1 E 0 (h, g) + E 1 (h, g)
where h = r 2 h + and g = r 2 g + for some defining function r, E m denotes any tensor whose components in any coordinate system smooth up to the boundary of the g, g -1 , h, h -1 and their partial derivatives such that in each term the total number of derivatives of g and h that appear is at most m. More precisely, we have E 0 (h, g) m = -h jk (g jk r m -4h mj r k ),

E 1 (h, g) m = -h jk ∂ k g mj -Γ(h) l mk g lj -Γ(h) l jk g lm + 1 2 ∂ m (h jk g jk )
If there is no confusion, we drop the index i for the metrics g + i , h + i , g * i , h * i , the covector field X i . In view of Theorem 4.4, we note B g + (h + ) = B g + (h + -g + ) ∈ C 1,α δ for all δ ∈ (0, 3). Moreover, B g + i (h + i ) → 0 in C 1,α δ . We consider a C 1 fully nonlinear operator Ψ for δ ∈ (2, 3)

Ψ : C 3,α δ (B 4 ; T B 4 ) → C 1,α δ (B 4 ; T B 4 ) X → B g + (exp( X) * h + )
where exp is the exponential map and B is a vector field related to the one form B. We know 2dΨ(0) = ∇ * ∇ + 3. It follows from Theorem C [START_REF] Lee | Fredholm operators and Einstein metrics on conformally compact manifolds[END_REF] that dΨ(0) : C 3,α δ (B 4 ; T B 4 ) → C 1,α δ (B 4 ; T B 4 ) is an isomorphism provided 2 < δ < 3. Applying inverse functions theorem, for large i, we find X i ∈ C 3,α δ such that Ψ( X i ) = 0.

Again from Lemma 3.7 [START_REF] Lee | Fredholm operators and Einstein metrics on conformally compact manifolds[END_REF], we have C 3,α δ (B 4 ; T B 4 ) ⊂ C 3,α 2+α (B 4 ; T B 4 ) ⊂ C 3,α (0) ( B4 ; T B4 ) provided δ > 2 + α. Thus, we find a diffeomorphism ϕ i = exp( X i ) of class C 3,α (equal to the identity on the boundary), such that F ((ϕ i ) * h + i , g + i ) = 0. Moreover, ϕ i (x) -x C 3,α → 0 and ϕ * i h + i -g + i C 2,α δ → 0 for δ ∈ (2 + α, 3). Thus we proved the claim.

Step 2. Claim. For large i, we have

g + i = ϕ * i h + i .
We know F (ϕ * i h + i , g + i ) = F (g + i , g + i ) = 0 and by step 1 g + i -ϕ * i h + (2 + α, 3). On the other hand, using [START_REF] Lee | Fredholm operators and Einstein metrics on conformally compact manifolds[END_REF]Theorems C and D] and [START_REF] Besse | Einstein manifolds[END_REF]Lemma 12.71], the linearized operator D 1 F (g + i , g + i ) : C 2,α δ (B 4 ; Σ 2 B 4 ) → C 0,α δ (B 4 ; Σ 2 B 4 ) is an isomorphism. Applying the implicit function theorem, we infer the claim.

Remark 5.1. Lemmas 1.6, 2.1, 2.4, 2.6, 3.1-3.3, 4.2 can be generalized in all dimensions in a similar way. So is Theorem 1.9 once the compactness result is established.

2 -

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Lemma 3 . 2 .

 32  Theorem 1.3]) Suppose that (X 4 , g + ) is a conformally compact Einstein manifold with the conformal infinity of Yamabe constant Y (∂X, [ĝ]) > 0. Then, for any p ∈ X 4 , (3.2)

  is the Euclidean half space (please see the details in the proof of [16, Proposition 4.8]). No interior blow-up: Next we consider the rest cases when dist ḡi (p i , ∂X i ) → ∞ 17 Jul 2020 01:43:24 PDT 190527-YuxinGe Version 2 -Submitted to Adv. Math.

i C 2 2 -

 22 Submitted to Adv. Math.

  light of(1.3). One may use Bach equations coupled with Q-flat equation to derive estimates for the Schouten tensor. To see Bach equations coupled with Q-flat equation also provide estimates of Weyl curvature, one may rewrite Bach equation as follows:

17 Jul 2020 01:43:24 PDT 190527-YuxinGe Version 2 -Submitted to Adv. Math.

the

2 .

 2 Lemma 2.3. ([16, Theorem 3.1 and Theorem 3.2]) Suppose that X is a smooth 4-manifold with boundary ∂X and g + is a conformally compact Einstein metric on X with the conformal infinity of positive Yamabe type. Let g * = ρ 2 g + be the Fefferman-Graham's compactification associated with the Yamabe metric of the conformal infinity. Assume the Sobolev inequalities (2.8) and (2.9) hold for the Fefferman-Graham's compactification g Sobolev inequalities. In the above estimates, Sobolev inequalities are essential. We need to control the constants C s and C b in Sobolev inequalities (2.8) and (2.9) in terms of other geometric quantities in Riemannian geometry. For the convenience of readers, we recall the following notions of injectivity radii for a Riemannian manifold (X, g) with boundary ∂X. For any interior point p ∈ X, let i int (p, g) be the supremum of r such that the normal geodesic γ(t) from p is minimizing for any t ∈ [0, min{r, t γ }],

	(2.12)
								1
	sup B(p,r/2)	|∇ k Rm| ≤	C k r k+2	B(p,r)	|Rm| 2 dvol[g * ] +	B(p,r)∩∂X	|S|dvol[ĝ] + vol(B(p, r))	2	.
	2.3.

* . Then there exists constants ε > 0 and C k > 0 such that if Rm L 2 (B(p,r)) ≤ ε 17 Jul 2020 01:43:24 PDT 190527-YuxinGe Version 2 -Submitted to Adv. Math. for a geodesic ball B(p, r) ⊂ X, then, for each k = 0, 1, 2, • • • ,

  there is a boundary harmonic coordinate chart on B(p, 4r) such that (2.13) and (2.14) hold in B(p, 2r).

17 Jul 2020 01:43:24 PDT 190527-YuxinGe Version 2 -Submitted to Adv. Math.

  4/3 . This yields that ρ ∞ (p) is bounded. This contradicts the claim in Step 3. Thus we have finished the proof of Lemma 4.2, hence the proof of Theorem 1.1.

17 Jul 2020 01:43:24 PDT 190527-YuxinGe Version 2 -Submitted to Adv. Math.

  For each Fefferman-Graham's compactification g * i (resp. h * i ), we first set the Fermi coordinate from the boundary. By the lower bound of the boundary injectivity radius, we know that the heights of these Fermi coordinates are bounded from the below. The necessary diffeomorphisms that fix the boundary one needs to use is to make sure that each of these Fefferman-Graham compactification g * i share the same distance function r to the boundary ∂X at least within the focal loci of g * i (resp. h * i ).

	boundary) that ψ * i g * i -φ * i h * i converges in weighted C 2,α δ denotes Fefferman-Graham's compactification of g + i (resp. h + topology, where g * i (resp. h * i ) i ).
	Proof.

  We denote ψ * i g + i by g + i and φ * i h + i by h + i . We denote Fefferman-Graham's compactification g * i = ρ 2 i g + i and h * i = ρ 2 i h + i where log ρ i and log ρ i solve (1.2). Let us consider the nonlinear functional on 4-dimensional ball B 4 introduced by Biquard [10] for two metrics g + and t + . (5.1) F (g + , t + ) := Ric[g + ] + 3g + -δ g + (B t + (g + )), where B t + (g + ) is a linear condition, essentially the infinitesimal version of the previous harmonicity condition B t + (g + ) := δ t + g + + 1 2 dtr t + (g + ).

Research of Chang is supported in part by NSF grant MPS-1509505. Research of Qing is supported in part by NSF grant MPS-1608782. 190527-YuxinGe Version 2 -Submitted to Adv. Math.


Keywords: seborrheic dermatitis, malassezia, child

de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

La dermatite séborrhéique de l'enfant

Seborrheic dermatitis in children

Mots-clés : dermatite séborréhique ; malassezia ; enfant

Introduction:

La dermatite séborrhéique [START_REF] Molinari | La dermatite séborrhéique: De la clinique au traitement[END_REF] est une affection moins fréquente chez l'enfant que chez l'adulte mais on note alors deux pics de fréquence, correspondant à deux formes cliniques différentes. Chez le nouveau-né et le nourrisson, elle est bipolaire ou diffuse, parfois impressionnante. Chez l'adolescent, elle est semblable à la forme de l'adulte, bénigne mais récidivante et souvent mal vécue du fait de ses localisations. Elle peut donc engendrer un fort retentissement psychosocial et une forte demande thérapeutique.

La prévalence de la dermatite séborrhéique est estimée entre 2 à 4 % [START_REF] Chosidow | Dermatite séborrhéique[END_REF]. Très fréquente dans les premiers mois de la vie, elle devient rarissime après 3 ans. Chez l'adulte, le pic de fréquence de la maladie se situe entre 18 et 40 ans avec une forte prédominance masculine, mais la maladie débute en général pendant l'adolescence.

Physiopathologie

La physiopathologie de la dermatite séborrhéique reste encore peu claire. La séborrhée interviendrait comme facteur favorisant le développement de ses levures.

En effet, les aires cutanées préférentielles de la maladie sont celles où les levures sont les plus nombreuses et où la sécrétion sébacée est la plus importante. De plus, la dermatite séborrhéique est plus fréquente aux âges de la vie où la sécrétion sébacée est plus importante et chez les individus de sexe masculin.

Le rôle des levures du genre Malassezia semble fondamental [START_REF] Quéreux | Dermatite séborrhéique[END_REF]. La dermatite séborrhéique n'est pas une maladie infectieuse mais plutôt une réaction inflammatoire de la peau aux levures car Malassezia est une levure saprophyte. Les traitements antifongiques sont efficaces mais ils ont aussi un rôle anti-inflammatoire. Malassezia a un rôle immunogène ou pro-inflammatoire (peroxydation de lipides) encore mal compris. Enfin, il existe des facteurs saisonniers (la dermatite séborrhéique étant moins fréquente en été) et le stress apparaît comme le principal facteur déclenchant des poussées [START_REF] Misery | Stress et dermatite séborrhéique[END_REF].

Diagnostic

La lésion élémentaire de dermatite séborrhéique associe un érythème à des squames plus ou moins grasses et épaisses. Les lésions sont souvent prurigineuses et s'accompagnent parfois de sensations de brûlure. Le diagnostic clinique est facile et en général suffisant.

L'examen anatomo-pathologique apporte des données peu spécifiques : acanthose hyperplasique de l'épiderme inter-folliculaire avec formation de petites squames parakératosiques souvent peri-ostiales, inflammation dermique superficielle, périvasculaire et périfolliculaire avec une exocytose lymphocytaire, légère spongiose de l'épithélium des infundibula pilaires et de l'épiderme autours des ostia pilaires hébergeant des levures de type Malassezia.

Chez le nouveau-né

Vers 3 à 4 semaines peut apparaître un érythème du visage et des croûtes au niveau du cuir chevelu. La dermatite peut devenir bipolaire, avec une atteinte du siège, ou plus diffuse réalisant une érythrodermie de Leiner -Moussous. Les filles et les garçons sont touchés de façon identique. Ces lésions rentrent habituellement dans l'ordre en quelques jours. Sans que l'on sache pourquoi, cette pathologie est beaucoup moins fréquente que dans le passé (lien avec l'hygiène ?).

Chez le nourrisson

Une éruption érythémato-squameuse ou croûteuse peut apparaître vers la fin du premier ou du deuxième mois de vie. Bien souvent, la dermatiite est bipolaire.

L'atteinte du siège peut prendre l'aspect de psoriasis avec des squames on parle alors de napkin psoriasis. Actuellement, une atteinte associée des grands plis, des creux axillaires, ou même une érythrodermie généralisée de Leiner-Moussous est plus rare.

En quelques semaines, les lésions disparaissent mais l'évolution peut être prolongée.

Le diagnostic différentiel se discute surtout avec la dermatite atopique, au cours de laquelle les lésions sont mal limitées.

Chez le grand enfant et l'adolescent

La dermatite séborrhéique est exceptionnelle jusqu'à la puberté. Chez l'adolescent, l'aspect est différent de la forme infantile de la maladie et devient proche de celle retrouvée chez l'adulte. Les lésions s'installent progressivement, débutant généralement par une atteinte du cuir chevelu. Elle évolue par poussées. L'atteinte du cuir chevelu est la plus fréquente (95 % des cas). Il s'agit de l'état pelliculaire ou pityriasis capitis. L'atteinte est assez diffuse mais forme des petites plaques au début. Elle siège préférentiellement sur les régions pariétales et le vertex, recouvert de fines squames. Elles débordent rarement sur le front ou les tempes mais souvent dans la région rétro-auriculaire ou la base du cou. Les lésions elles-mêmes sont donc en général masquées par les cheveux mais les pellicules sont très visibles sur les cheveux ou sur les vêtements.

Le visage est la 2 e localisation de la dermatite séborrhéique (65 % des patients). La localisation des lésions est médio-faciale : sillons naso-géniens, glabelle, partie médiane des sourcils, lisière du front, pli sous-labial. Assez souvent, la barbe et la moustache sont atteintes, plus rarement les joues, presque jamais le dorsum nasal et la pointe du nez. L'atteinte des cils peut entraîner des blépharites et des conjonctivites.

Le tronc représente la 3 e localisation de la dermatite séborrhéique (30 % des patients).

La topographie est caractéristique : pré-sternale, plus rarement inter-scapulaire. Les lésions débutent par des macules péri-pilaires puis évoluent vers des médaillons ronds, ovalaires ou circulaires. Elles sont parfois jaunâtres ou un peu orangées. On constate fréquemment l'effet bénéfique de l'ensoleillement sur la dermatite séborrhéique.

La dermatite séborrhéique du nourrisson ne nécessite souvent pas de traitement spécifique mis à part des soins d'hygiène habituels avec un syndet en évitant l'utilisation de produits détergents qui sont irritants. Sinon, les mêmes traitements que ceux utilisés chez l'adulte peuvent être utilisés.

1 Résumé

 1 : laurent.misery@chu-brest.fr Liens d'intérêt : Beiersdorf, Bioderma, Expanscience, Johnson&Johnson, Leo Pharma, L'Oréal, Nestlé, Pierre Fabre, Uriage La dermatite séborrhéique est une dermatose érythémato-squameuse concernant essentiellement le cuir chevelu (état pelliculaire) et le visage mais pouvant avoir d'autres localisations et être assez diffuse chez l'enfant. La physiopathogénie est discutée mais le rôle de Malassezia semble important. La dermatite séborrhéique du nouveau-né et du nourrisson, bipolaire, est assez fréquente. La dermatite séborrhéique devient ensuite très rare pour devenir très fréquente à l'adolescence. Mise à part la forme clinique spécifique au nourrisson, l'évolution est chronique et récidivante. Le traitement est essentiellement topique et basé surtout sur les antifongiques ou le gluconate de lithium.

Traitement [ 6 , 7 ]

 67 Les autres localisations sont bien plus rares : grands plis (patients obèses ou immunodéprimés), organes génitaux (surtout le gland et le sillon balano-préputial), conduit auditif externe. Une atteinte diffuse et/ou très inflammatoire d'installation rapide doit faire rechercher une infection à VIH chez l'adolescent [5]. Le diagnostic différentiel se discute essentiellement avec le psoriasis, surtout s'il n'y a pas d'autre lésion sur le corps. Les lésions psoriasiques sont plus sèches, plus épaisses, mieux limitées et plus érythémateuses et les squames sont plus brillantes et moins grasses. Les antifongiques sont le principal traitement de la dermatite séborrhéique. Les imidazolés, dont le kétoconazole, sont les plus fréquemment utilisés. On les utilise sous forme de gel moussant ou de crème. Un traitement d'attaque à deux applications par semaine puis un traitement d'entretien à une application par semaine est souvent proposé. D'autres imidazolés, comme le fluconazole, le miconazole, le bifonazole, ou le métronidazole peuvent aussi être utilisés, ainsi que la ciclopiroxolamine ou la terbinafineet des shampooings au pyrithione-zinc ou à la piroctone-olamine. Il faut noter que le propylène glycol souvent présent dans les shampooings a en lui-même une activité importante dans la dermatite séborrhéique. Le gluconate de lithium inhibe la synthèse d'acides gras et a une activité anti inflammatoire. Il est disponible sous forme de crème à 8 %. Les dermocorticoïdes ont un effet rapide mais leur utilisation doit être limitée du fait des effets rebond et de leurs effets secondaires. Le tacrolimus a montré une certaine efficacité mais la seule indication faisant l'objet d'une A.M.M. reste la dermatite atopique. L'acide salicylique a une action kératolytique. Il est souvent associé à d'autres molécules dans des produits cosmétologiques tels que les shampooings antipelliculaires à des concentrations inférieures à 2 %. Le sulfure de sélénium est antifongique, kératolytique et anti-séborrhéique. Il est présent dans des shampooings mais est moins utilisé actuellement, probablement du fait d'une odeur désagréable. Les traitements systémiques (itraconazole, fluconazole, terbinafine) sont réservés à des très rares formes particulièrement invalidantes et profuses, résistant aux traitements topiques.

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