Conformally Compact Einstein Metrics with Symmetry on 5-Manifolds Mohammad Javaheri January 13, 2007 Abstract We study the Dirichlet problem for conformally compact Einstein metrics on 5-manifolds with globally static isometric circle actions. As an application of our general results, we show that any non-flat analytic warped product metric on ∂M ×S 1 with non-negative scalar curvature is the conformal infinity of some Einstein metric on M 4 × S 1 .

Contents 0 Introduction

1

1 Background Results

5

2 The Banach Manifold EG,AH

7

3 The Interior Behavior

11

4 The Boundary Behavior

23

References

32

0

Introduction

Conformally compact Einstein metrics have recently been of great interest to physicists and mathematicians alike. Understanding the interesting relationship between the geometry of the interior metric of a conformally compact 1

Einstein metric and its conformal boundary is a major issue in the AdS/CFT correspondence, a field that is under intense investigation by string theorists; see Witten [21, 1998] for instance. The mathematical work in this area started with the works of Fefferman and Graham in [10, 1985] who studied the conformal invariants of Riemannian manifolds. ¯ = N ∪ ∂N be a compact oriented and connected smooth manifold Let N with boundary ∂N . A defining function for N is a non-negative C ∞ function ¯ such that ρ−1 (0) = ∂N and dρ 6= 0 on ∂N . (N, g) is called defined on N conformally compact if there exists a defining function ρ such that g¯ = ρ2 g extends continuously to the boundary. This definition is essentially due to Penrose. If a C m,α extension to the boundary exists, g is called C m,α conformally compact. In this paper we only consider metrics which are at least C 2 conformally compact. Einstein metrics by definition are metrics that satisfy the Einstein equations Ricg = Λg for some constant Λ. A metric g on N n+1 is called AH or asymptotically hyperbolic if its sectional curvature approaches -1 as one approaches ∂N . If an Einstein metric g is AH and conformally compact, then one has Ricg = −ng , (0.1) and g is said to be AHE. m,α Let EAH denote the set of conformally compact AHE metrics g on N for which there exists a defining function ρ such that the induced boundary metric γ = ρ2 g|∂N is C m,α on ∂N . The group D1 of C m+1,α diffeomorphisms of N which induce identity on ∂N acts on this set by pull-back, and so one can form the moduli space (cf. §1 for more details) m,α m,α EAH = EAH /D1 .

(0.2)

These moduli spaces are studied in [2] to approach the Dirichlet problem for this class of metrics, i.e. to find an Einstein metric with a given conformal infinity. This is a difficult analytic problem, since the equations form a non-linear elliptic system which degenerates on the boundary. Using inverse function theory arguments, Graham and Lee [13, 1991] proved that if the conformal structure is close enough to that of the round sphere, then an Einstein filling exists, which is unique among Einstein metrics close to the Poincar´e metric. Biquard [7, 2000] generalized this result to arbitrary nondegenerate Einstein manifolds.

2

On the other hand, uniqueness for the Dirichlet problem fails in general; this was first observed by Hawking and Page [14, 1983]; cf. also [1] for some examples and discussions. A more global existence result holds in cases where the boundary metric is conformal to a non-flat metric with non-negative scalar curvature. Consider the boundary map 0 → C 0 (∂M ), Π0 [g] = γ = g|∂N , Π0 : EAH

(0.3)

where C 0 is the connected component of the round metric in the set of conformal classes of non-flat metrics with non-negative scalar curvature on the 0 = (Π0 )−1 C 0 . In [2, 2001], it is proved that, in dimension boundary, and EAH 4, under a mild topological assumption, the map above is proper and, for example on (B4 , S 3 ), it is surjective. The translation of this statement is that any non-flat metric on S 3 with non-negative scalar curvature, whose conformal class is connected to that of the standard round metric, can be realized as the conformal boundary of a unique conformally compact Einstein metric in the connected component of the standard Hyperbolic metric on B4 . In this paper, we consider complete conformally compact Einstein metrics with isometric group actions. Suppose a group G acts on N and consider the space of conformally compact Einstein metrics on N which are G-invariant and induce G-invariant C m,α boundary metrics on ∂N . We denote this space m,α m,α , the moduli space of such metrics under . First we consider EG,AH by EG,AH diffeomorphisms that fix the boundary and preserve the G-action. We show that, if non-empty, this is a separable C ∞ Banach manifold and the boundary map, defined in a similar manner as before, is Fredholm (cf. §2). After this general result is proved, we focus on strictly globally static circle actions on N 5 i.e. metrics of the form N = M × S 1 , g = h + u2 dθ2 ,

(0.4)

where u is a smooth function on N with u > 0 and h is a smooth metric on M . Anderson, Chru´sciel and Delay [5] have studied static circle actions in connection to Lorentzian vacuum solutions of Einstein equations and have proved several local existence results. The Einstein equations (0.1) on N = M × S 1 descend to the following equations on M  RicM = −4h + u1 D2 u , (0.5) ∆u = 4u , 3

where D2 u denotes the Hessian of u. These are the Einstein equations on M coupled with a scalar field. These equations are used in § 3 to study the interior behavior of the metric. The following theorem is the main result of this paper. Let Cω0 be a connected component of conformal classes of real analytic metrics on the boundary which contain a non-flat metric with non-negative scalar curva0 = ES01 ,AH = Π−1 Cω0 . ture, and let E1,AH ¯ Theorem A. Let N 5 = M 4 × S 1 and suppose the inclusion i : ∂M → M induces a surjection ¯ , R) → 0 . H2 (∂M, R) → H2 (M

(0.6)

0 Then the boundary map Π0 : E1,AH → Cω0 is proper.

As an application of this theorem, we prove that, for N = R5 /Z ' R4 ×S 1 , the boundary map above is surjective. Here the Z quotient comes from a translation of length L along a geodesic in H5 (−1), the standard hyperbolic space of dimension 5 with constant sectional curvature K = −1. This gives a conformally compact Einstein metric g0 with conformal infinity equal to S 3 × S 1 (L) with the standard product metric. The idea is to show that the Z-valued Anderson degree of the boundary map is non-zero. The Z-valued Anderson degree of the boundary map above is defined as follows. Consider the elliptic operator 1 ˆ, L = D∗ D − R 2

(0.7)

ˆ is the action of the curvature operator R on acting on L2,2 (N, g), where R 2-tensors given by (Rˆg h)(X, Y ) =

n+1 X

h(R(X, ei )ei , Y ) ,

(0.8)

i=1

where ei form an orthonormal basis. Let indg ∈ Z be the maximal dimension of the subspace of L2,2 (N, g) on which L is negative definite with respect to the L2 inner product. Then deg Π0 is defined as X (0.9) deg Π0 = (−1)indgi , gi ∈(Π0 )−1 ([γ])

4

where [γ] is a regular value of Π0 . It is shown in [2, §5] that this degree is well-defined and independent of the choice of the regular value [γ]. Theorem B. For N = B5 /Z and ∂N = S 3 × S 1 as above, one has: deg Π0 = 1 .

(0.10)

In particular any conformal class [γ] ∈ Cω0 on S 3 × S 1 in the connected component of the standard product metric is the conformal infinity of some conformally compact Einstein metric on B4 × S 1 (that is in the connected component of the standard hyperbolic metric on N).

Acknowledgment I would like to thank Michael Anderson for his comments and generous help.

1

Background Results

Throughout this section, choose a fixed defining function ρ0 on N . For positive integer k and 0 < β < 1, let Sk,β be the Banach space of symmetric 2 bilinear forms h on N such that kρ−2 0 hkC k,β < C ,

(1.1)

for some constant C < ∞, where the above norm is the usual C k,β norm with respect to g. For any boundary metric γ, we define a standard corresponding AH metric gγ on N that induces γ on the boundary as follows. Fix a collar neighborhood U of ∂N and identify U with [0, 1] × ∂N in such a way that the first component is given by ρ0 . For r = − log(ρ0 /2) let gU = dr2 + sinh2 (r) · γ .

(1.2)

Next let U 0 be a thickening of U and let η be a function with the following properties: η = 1 on U , η = 0 on M \ U 0 and dη 6= 0 on M \ U 0 . Finally define gγ = (1 − η) · gC + η · gU , (1.3) where gC is a fixed smooth Riemannian metric on M \ U .

5

Definition 1. Let M etk,β AH denote the set of all metrics on N of the form gγ + h such that γ ∈ M etm,α (∂N ) and h ∈ Sk,β 2 . Define k,β EAH ⊂ M etk,β AH ,

(1.4)

as the subset of AH Einstein metrics g with Ricg = −ng. In addition let m,α k,β EAH be that subset of EAH comprised of metrics which induce C m,α metrics on the boundary. Any defining function induces a boundary metric γ∂N on ∂N . Conversely, associated to any boundary metric γ∂N , there exists a unique geodesic defining function ρ, defined in a neighborhood of the boundary, cf. [13, Lemma 5.2], such that g¯ = ρ2 g induces γ∂N and ¯ g¯ = 1. |∇ρ|

(1.5)

¯ and the boundary metric γ∂N is invariant If a compact group G acts on N under this group action, then the geodesic defining function associated with γ∂N is invariant under the group action too (by uniqueness), and so it is a geodesic defining function on the quotient space as well. This naturally gives k,β rise to the definition of M etk,β G,AH and EG,AH ; cf. § 2 for more discussions. The action of diffeomorphism groups. k,β There are two diffeomorphism groups acting on EAH that are of interest to us. These groups are D1 , the group of orientation preserving C m+1,α diffeomorphisms that induce identity on the boundary and D2 , the set of φ ∈ D1 that satisfy φ∗ ρ0 lim ( ) = 1. (1.6) ρ0 →0 ρ0 The group D2 is a normal subgroup of D1 with the quotient D1 /D2 which is naturally isometric to the group of C m,α positive functions on ∂N . The k,β action of the group Di on EAH is free and proper. It follows that (i)

k,β EAH = EAH /Di ,

(1.7)

are C ∞ separable Banach manifolds (i = 1, 2), with boundary maps: (2)

(1.8)

(1)

(1.9)

Π : EAH → M etm,α (∂N ), Π[g] = γ, Π : EAH → C m,α , Π[g] = [γ], 6

where C m,α = C is the Banach space of C m,α conformally equivalent metrics on the boundary. These boundary maps are C ∞ and Fredholm of index zero [4]. The following results are from [2]. Let C 0 ⊂ C denote the subset of nonnegative conformal classes i.e. classes which contain some non-flat metric 0 = Π−1 (C 0 ). Let M be a 4with non-negative scalar curvature, and EAH manifold such that the inclusion i : ∂M → M induces a surjection ¯ , R) → 0. H2 (∂M, R) → H2 (M

(1.10)

0 Then for any (m, α), m ≥ 4, the boundary map Π0 : EAH → C 0 is proper. 0 Given this, an integer-valued degree on each component of EAH is defined, cf. eq. (0.9). Existence results can be established by showing that this degree is nonzero. For example, in case of M = B4 , the four dimensional ball, the degree of the boundary map on the component including the hyperbolic metric is shown to be 1, cf. [2, Theorem B]; this implies that any conformal class [γ] ∈ C 0 on S 3 is the conformal infinity of some AH Einstein metric on B4 .

2

The Banach Manifold EG,AH

¯ = N ∪ ∂N . Throughout this Assume that a compact group G acts on N k,β paper, this group action is fixed. Let EG,AH be the G-invariant elements of k,β EAH , i.e. k,β k,β g ∈ EG,AH ⇔ g ∈ EAH ∧ (∀f ∈ G : f ∗ g = g). (2.1) In a similar manner, we define M etk,β G,AH , the set of G-invariant AH metk,β rics, and SG,2 , the G-invariant subset of Sk,β 2 . Next choose a background k,β G-invariant metric g0 ∈ EG,AH with a G-invariant boundary metric γ0 . For γ ∈ M etm,α G,AH (∂N ) close to γ0 , define g(γ) = g0 + η(gγ − g0 ) ,

(2.2)

where η is the G-invariant cutoff function in (1.3). Now define the Bianchigauged Einstein operator at g0 by m,α Φ : M etm,α (N ) → Sm−2,α (N ), 2 AH × S Φ(γ, h) = Φ(gγ + h) = Ricg + ng + (δg )∗ Bg(γ) (g),

7

(2.3) (2.4)

where Bg(γ) is the Bianchi operator Bg(γ) (g) = δg(γ) g + 21 d(trg(γ) g). It is proved in [7] that k,β k,β ZAH ≡ Φ−1 (0) ∩ {Ric < 0} ⊂ EAH ,

(2.5)

k,β and that ZAH provides a local slice to the action of the diffeomorphism group m,α near g0 ∈ EAH . The derivative of this map at g0 with respect to the second factor is the linearized Einstein operator in eq. (0.7). Related to our disk,β k,β cussion on group actions, let ZG,AH be the G-invariant subset of ZAH . We first show that the action of G on N induces an action of G on ZAH . To see this, notice that since g0 is Einstein, ZAH provides a slice to the action of diffeomorphism group on EAH through g0 , and since g0 is a fixed point of the action of G, this slice must be invariant under the G-action. We know that k,β k,β ZAH is a C ∞ Banach manifold [4] while ZG,AH is the fixed point set of the k,β compact group action G. It follows that ZG,AH is a C ∞ Banach submanifold k,β of ZAH . k,β k,β The set EAH differs from ZAH by the action of diffeomorphism group, in k,β that, for any g ∈ EAH near g0 , there exists a diffeomorphism φ such that k,β k,β . . We now prove a similar property for ZG,AH φ∗ g ∈ ZAH k,β k,β Lemma 2. Suppose g0 ∈ EG,AH is near g0 , , Isom(g0 ) = G and g ∈ EG,AH k,β ∗ then there exists a G-invariant diffeomorphism φ such that φ g ∈ ZG,AH . k,β Proof. Let φ∗ g = h ∈ ZAH . For any a ∈ G we have

(φ−1 aφ)∗ (h) = h .

(2.6)

Since Isom(g0 ) = G and φ−1 aφ fixes an element of the slice, we have φ−1 aφ ∈ G. It follows that the homomorphism φˆ : G → G defined by ˆ φ(a) = φ−1 aφ ,

(2.7)

is well-defined, hence one to one and onto. Then eq. (2.6) implies that h is a fixed point of the action. On the other hand, eq. (2.7) implies that φ is in N (G), the normalizer of G in the diffeomorphism group and so, ∀a ∈ G and for any G-invariant metric g, we have a∗ (φ∗ g) = (φa)∗ g = (bφ)∗ g = φ∗ (b∗ g) = φ∗ g ,

(2.8)

where b = φ−1 aφ ∈ G. This means that φ∗ g is G-invariant, and so φ is G-invariant by definition. 8

k,β The group D2 acts on EAH and preserves the boundary map in the sense that ∀φ ∈ D2 : Π(g) = Π(φ∗ g) . (2.9)

With regard to the group action G, we define D2G to be the G-invariant k,β subgroup of D2 . This group acts on EG,AH and the discussion above implies that the quotient (2) k,β (2.10) /D2G , EG,AH = EG,AH is a C ∞ separable Banach manifold, in the condition that there exists a g0 k,β in this set with Isom(g0 ) = G. This condition holds, if EG,AH is non-empty, k,β since the set of such elements are dense and open in EG,AH . Because of (2.9), the boundary map descends to a map (2)

Π : EG,AH → M etm,α G (∂N ), Π([g]) = γ .

(2.11)

The above discussion leads to the following: (2)

Proposition 3. The space EG,AH is a smooth separable Banach manifold, (if (2) ∞ non-empty). The boundary map Π : EG,AH → M etm,α Fredholm G (∂N ) is a C map of index 0, and Ker(DΠ)g = KgG , (2.12) where KgG is the space of L2 infinitesimal G-invariant Einstein deformations at g. Proof. The proof is essentially the same as the proof of [3, Prop. 4.3]; one only needs to replace every set by its G-invariant subset and apply the above discussion. Now let D1G be the G-invariant subgroup of D1 . These are C m+1,α and Ginvariant diffeomorphisms that induce identity on the boundary. It’s easy to see that D2G is a normal subgroup of D1G with the quotient D1G /D2G naturally isometric to the positive C m,α G-invariant functions on ∂N . Thus define (2)

(1)

k,β EG,AH = EG,AH /D1G = EG,AH /(D1G /D2G ) .

(2.13)

It’s straightforward to show that this is a C ∞ Banach manifold (if nonempty) and the boundary map Π naturally descends to a C ∞ boundary map Π : (1) EG,AH → CG which is Fredholm, of index 0 with kernel as in (2.12). 9

Boundary regularity. In dimension 4, it is shown in [1] that if g has an L2,p conformal compactification, for some p > 4, then it admits a compactification, with the same boundary metric, which is as smooth as the boundary metric. In the notation of §1, this says k,β m,α EAH = EAH . (2.14) However, in dimension 5, this result is definitely false because in the asymptotic expansion of the metric, log terms appear from the order 4 on, g¯ ∼ g(0) + ρ2 g(2) + ρ4 (log ρ)h(4) + ρ4 g(4) + . . .

(2.15)

While g(2) , h(4) , divergence δg(4) and T r(g(4) ) are all determined by g(0) = γ∂N , the transverse-traceless part of g(4) is not determined by the boundary metric. At the same time, all other terms in the formal expansion above are determined by g(0) and g(4) . The log terms that appear in the above expansion force the metric to be at most C 3,α conformally compact. On the positive side, the recent results of Chru´sciel, Delay, Lee and Skinner [9, 2004] show that for smooth boundary metrics, the polyhomogeneous expansion above exists to all orders. Also S. Kichenassamy [16, 2004] proves that, in case of analytic boundary metrics, the expansion (2.15) converges to g¯(x, ρ) and g¯(x, ρ) is analytic in ρ and ρ log ρ, where these two are considered as independent parameters. The result in [9] allows us to define E˜AH , the classes of metrics g which are conformally compact with a compactification g¯ which is a smooth function of (t, t4 log t, y) where t is the geodesic defining function and y parameterizes the boundary. Theorem 5.6 in [3] indicates that E˜AH is a smooth infinite dimensional Fr´echet manifold and Π : E˜AH → C ∞ is a C ∞ Fredholm map of index 0. The same proof that led to Prop. 3 can be carried over to the current discussion to prove the following: Proposition 4. Let E˜G,AH be the G-invariant subset of E˜AH . Then, if nonempty, this is a C ∞ infinite dimensional Fr´echet manifold with the boundary map (2.16) Π : E˜G,AH → CG∞ , where CG∞ is the G-invariant subset of C ∞ . The boundary map is smooth and Fredholm of index 0.

10

3

The Interior Behavior

In this section, we study the behavior of AH Einstein metrics in the interior (i.e. away from the boundary) of 5-manifolds with static S 1 actions. The key step is to control the L2 norm of the curvature tensor on large balls. Since for W , the Weyl curvature, and z, the trace-free Ricci, we have |R|2 = |W |2 + 12 |z|2 + 6, we need a control on the L2 norm of W and z, and a control on the volume of balls in M . These bounds are obtained in Lemma 6, Proposition 8 and Corollary 9. In the sequel, (N, g) is an AHE metric with a globally static circle action as in (0.4) and ρ is the geodesic compactification associated with the S 1 invariant boundary metric γ. Throughout this section, until Proposition 12, we assume that u > 0 on M . Here and throughout let BxM (R) = Bx (R), respectively BxN (R), be the ball of radius R centered at x in M , respectively N. The next Lemma is useful in controlling u and |∇u| on large domains in M . Choose a point y ∈ M where the minimum of u occurs and set m = u(y) > 0. Different choices for y are possible but we fix a y once for all. Lemma 5. Suppose (N 5 , g) is Einstein with Ricg = −4g and N = M × S 1 such that g = h + u2 dθ2 . Then there exists a constant α0 > 0 such that ∀x ∈ M u(x) |∇u(x)| 6 α0 u(x) · (log + 1) . (3.1) m Moreover, there exists a function G = G(R), defined for R > 0, such that ∀x ∈ By (R) u(x) 6 m · G(R) , |∇u(x)| 6 m · G(R) .

(3.2) (3.3)

Proof. Define

u(x) . (3.4) m Since u(x) > m, we have f (x) > 0 and f (y) = 0. It follows from eq. (0.5) that ∆M u =4. (3.5) ∆N f = u Following [8, Theorem 6], we conclude that ∀x ∈ M f (x) = log

|∇f (x)| 6 α0 (f (x) + 1) , 11

(3.6)

where α0 is a constant. This is the same as (3.1). To prove the second inequality, rewrite the above inequality as |∇{log(f (x) + 1)}| 6 α0 .

(3.7)

which implies that log(f (x) + 1) 6 α0 · distg (x, y) 6 Rα0 and so ∀x ∈ By (R) :

u(x) 6 m · exp ◦ exp(Rα0 ) .

(3.8)

Finally this and (3.1) give the upper bound on |∇u|. For future purposes, we need a volume comparison result which enables us to compare the volume of balls of different radii in M . Such a result holds on (N, g) by the usual Bishop-Gromov volume comparison theorem [19], since (N, g) is Einstein. Hence to obtain a similar result on (M, h), one only needs to compare the volume of balls in M with the volume of balls in N ; this is done in the following Lemma. Lemma 6. There exists a function V (r), r > 0 such that ∀x ∈ ByM (R) :

volBxN (r) 6 volBxM (r) 6 V (r) , 2r · G(R + r)

(3.9)

where G(R) comes from Lemma 5. Moreover for a > b > 0, there exists a constant c=c(a,b,R) such that ∀x ∈ ByM (R) : volBxM (a) 6 c · volBxM (b) .

(3.10)

Proof. Let η(t) be a geodesic in BxN (r) of speed one starting at x, and write η(t) = (φ(t), θ(t)), where the curves φ(t) and θ(t) are in M and S 1 respectively. Since |η 0 (t)|2 = |φ0 (t)|2 + u2 (φ(t))|θ0 (t)|2 = 1, we have |φ0 (t)| 6 1 and |θ0 (t)| 6 1/u(φ(t)). It follows that the length of the curve φ(t) is at most r; furthermore, since u(φ(t)) > m, we have Z r r dt 6 . (3.11) |θ(r)| 6 m 0 u(φ(t)) This means that BxN (r) ⊂ {(z, θ) ∈ M × S 1 : z ∈ BxM (r), |θ| 6 r/m} and so Z r N volBx (r) 6 2 (3.12) · udVg 6 2rG · volBxM (r) , m M Bx (r) 12

where G = G(R + r) comes from Lemma 5. This proves one part of (3.9). To prove the other inequalities, without loss of generality, by re-scaling u, 1 assume m = 1 and notice that {(z, θ) ∈ BxM (r) × S 1 : |θ| 6 u(z) } ⊆ BxN (r + 1) and so volBxM (r) 6 volBxN (r + 1) . (3.13) Now since (N, g) is Einstein with Ricg = −ng, one has volBxN (r + 1) 6 V (r), where V depends only on r. This proves the second half of (3.9). Then (3.10) follows from (3.12),(3.13) and the volume comparison theorem on (N, g). Next we need to prove a non-collapsing result. Lemma 7. Let N = M × S 1 be any AH Einstein manifold of dimension 5 as in (0.4), with C 2,α geodesic compactification g¯ = t2 · g and boundary metric γN . In addition assume that ¯ ∂N ) > τ¯ > 0 , Ing¯(∂N ) = distg¯(C,

(3.14)

where C¯ is the cutlocus of the boundary, and diamg¯SN (t1 ) 6 T ,

(3.15)

for t1 = τ¯/2 and SN (t1 ) = {x ∈ (N, g¯) : t(x) = t1 }. Then for all x ∈ M with d 6 distg¯(x, ∂N ) 6 D ,

(3.16)

volBxM (1) > ν0 > 0 ,

(3.17)

we have where ν0 depends on (∂N, γN ), d, D, T, τ¯ and distg (x, y). Proof. Define E¯N (x, t) to be the inward exponential map of (N, g¯) at ∂N and J¯N (x, t) to be its Jacobian. Also let τ0 (x) be the distance to the cutlocus of E¯N at x ∈ ∂N . Then by using the infinitesimal Bishop-Gromov volume comparison theorem, we conclude that J¯N (x, t) ↑ τ04 (1 − ( τt0 )2 )4

(3.18)

is monotone non-decreasing in t, for any fixed x. From this, it’s straightforward to show that volg¯SN (t) ↑ (3.19) 4 τ¯ (1 − ( τ¯t )2 )4 13

By taking t = t1 = τ¯/2, this implies that volg¯SN (t1 ) > (3/4)4 volγN ∂N, or equivalently volN SN (t1 ) > C , (3.20) where C depends on γN and t1 . Since N is Einstein, there is c(R) such that for R > 1 : volBxN (1) > c(R) · volBxN (R). Conditions (3.15) and (3.16) imply that there exists R such that SN (t1 ) ⊂ BxN (R − 1) and so: volBxN (1) > c(R) · volBxN (R) > c(R) · vol(BxN (R)\BxN (R − 2)) 1 1 > c(R) · volSN (t1 ) > c(R) · C = ν0 > 0. (3.21) 2 2 Now the result follows from (3.21) and (3.9). We now prove an upper bound on the L2 norm of the trace-free Ricci. In order to proceed, we impose the following condition on the compactified metric g¯ near ∂N ; this condition will be removed in the next section (through Prop. 14 and Cor. 17). Boundary Condition (BC). There exists a geodesic compactification g¯ = ρ2 g, and constants ρ0 , C > 0 such that for L = {x ∈ N | ρ(x) 6 ρ0 }, the C 2,α norm of g¯ (with respect to a fixed background metric) is bounded by C for some α ∈ (0, 1). Notice that (BC) implies the bounds (3.14) and (3.15), and so (3.17) is valid for x ∈ M with ν0 depending on bounds on distg¯(x, ∂N ) and distg (x, y). From now until § 4, we enforce (BC) and all of the constants that appear in our estimates automatically depend on C, ρ0 and α as well as other quantities that we will specify. However, to be more accurate, we use the phrase ’depending on (BC)’ by which we mean that first of all (BC) is enforced and secondly the constant under consideration depends only on C, ρ0 , α and possibly other constants that we will specify. In the following Proposition, the width of (N, g¯) is the length of the longest ρ-geodesic from ∂N to points in the interior i.e. W idg¯ = sup{distg¯(x, x1 )| x ∈ N , x1 ∈ ∂N } .

(3.22)

Proposition 8. Suppose that for some ω0 < ∞, W idg¯ 6 ω0 . 14

(3.23)

Then for any ρ > 0 there exists a constant K = K(ρ, ω0 ) depending only on (BC), ρ and ω0 such that for the trace-free Ricci curvature z Z |z|2 dvolg 6 K(ρ, ω0 ) , (3.24) B(ρ)

where B(ρ) = {x ∈ M | ρ(x) > ρ}. Proof. Let S(ρ) be the set of points x ∈ M such that ρ(x) = ρ. Recall that uz = D2 u − uh and trh z = hz, hi = 0. One has u|z|2 = hz, uzi = hz, D2 ui = δ{z(du, .)} − hδz, ∇ui ,

(3.25)

where z(du, .) is viewed as a 1-form and δ denotes the divergence. Since δz = δRic = 12 dsg = 0, divergence theorem implies that Z Z Z 2 u|z| dvolg = δ{z(du, .)}dvolg = z(du, dr)dvolg . (3.26) B(ρ)

B(ρ)

S(ρ)

Without loss of generality we assume ρ 6 ρ0 /2, where ρ0 is provided by (BC). Since ρ is the geodesic defining function, equation (0.5) gives rise to ¯ = ρ2 h), [6]: the following equation on (M, h ¯ 2ρ ¯ D ∆ρ ¯+z . ¯ Ric = −2 −( )h ρ ρ

(3.27)

This implies that on S(ρ) ¯2 ¯ ¯ ¯ + 2 D ρ + ( ∆ρ )h}(du, z(du, dr) 6 |{Ric dr)| 6 K1 · ρ|du| , ρ ρ

(3.28)

where K1 is independent of ρ (although it depends on C, ρ0 , α from (BC)). Also volS(ρ) = ρ−3 volh¯ S(ρ) 6 ρ−3 v1 , for a constant v1 independent of ρ. From (3.26) and (3.28) one has Z Z K1 v1 K1 v1 2 m |z| 6 u|z|2 6 2 · max |du| 6 2 · m · G(R) , (3.29) ρ ρ B(ρ) B(ρ) where G(R) comes from Lemma 5 and R = max dist(y, x) for x ∈ B(ρ). It is left to show that R is bounded from above. Choose y1 on the intersection 15

of the geodesic connecting y to ∂N in (N, g¯) and S(ρ0 /2). Then for any x ∈ B(ρ) we have dist(y, x) 6 dist(y, y1 ) + dist(y1 , x) 6

ω0 − ρ0 /2 2 + diamg¯S(ρ0 /2) . (3.30) ρ0 /2 ρ0

This gives an upper bound for R and the proposition follows. Proposition 8 leads to the following corollary which provides a L2 control on the Weyl curvature. Corollary 9. Suppose W idg¯ ≤ ω0 < ∞. Then there is a constant λ0 < ∞, that depends only on (BC) and ω0 such that Z |W |2 dvolg 6 λ0 < ∞ , (3.31) M

Proof. The idea is to find two bounds, one bound on the domain Ω = B(ρ0 /2) and another bound on the complement Ωc = M \B(ρ0 /2). R In the region c ¯ 2 Ω R the2 Boundary Condition provides an upper bound for |W | dvolg¯ = |W | dvolg , while on Ω, we use the Chern-Gauss-Bonnet theorem: Z Z Z 1 2 2 2 |W | = 8π χ(Ω) − 6volΩ + |z| + B(R, A) , (3.32) Ω Ω 2 ∂Ω where B(R, A) is some boundary term that depends on the curvature R and A, the second fundamental form of S(ρ0 /2). Again by Boundary Condition this boundary term and vol∂Ω are uniformly controlled, and by Proposition 8, the term involving z is also controlled. This proves the assertion. R One may attempt to find an intrinsic upper bound for M u|z|2 , i.e. an upper bound that depends only on the conformal boundary, and from there R 2 conclude a bound for M |z| . The example below shows that, in general, this integral is not intrinsic to the conformal boundary. Example. Consider N = R2 × S 3 with the AdS Schwarzschild metric: g = V −1 dr2 + V dθ2 + r2 gS 3 (1) , 2m V = 1 + r2 − 2 . r 16

(3.33) (3.34)

If we denote the largest zero of V by r0 , (3.34) gives the metric on the domain (r0 , ∞) × S 1 × S 3 . We define M = (r0 , ∞) × S 3 , where ∂M = S 3 (1) and ∂N = S 1 (β) × S 3 (1). Here u = V 1/2 gives the length of the orbits and a simple calculation yields D2 (ei , ej ) = 0, D2 u(ei , ei ) = u(1 + D2 u(∇r, ∇r) = u(1 −

2m ), r4

6m −1 )V , D2 u(∇r, Xi ) = 0, r4

(3.35) (3.36)

where ei make an orthonormal frame for g. One has ∀X, Y ⊥∇r : z(X, Y ) =

2m 6m g(X, Y ), z(∇r, ∇r) = − g(∇r, ∇r), (3.37) r4 r4 2

which implies that |z|2 = 48 mr4 and so Z Z ∞Z m2 2 u|z| = V 1/2 (48 4 )V −1/2 r3 drdµ = 12m2 vol(S 3 )/r04 r M r0 S3

(3.38)

We notice that ∂N = S 1 (β) × S 3 (1), where β = 4πr0 /(2 + 4r02 ); it is easy to see that two values of m may define the same conformal boundary (i.e. the same β) while the above mentioned integrals have different values. To be able to use Lemma 7 we need to have (3.16) which gives a bound on the distance of the base point from the boundary in the compactified metric. Lemma 10. Suppose y is a point where the minimum of u occurs. Then there exists ρ1 > 0 depending only on (BC) such that distg¯(y, ∂N ) > ρ1 .

(3.39)

Proof. A simple calculation shows ∂ (ρu) = ρ2 u(¯ s/6 − s¯N /8) , ∂ρ

(3.40)

¯ and (N, g¯) respectively. where s¯ and s¯N are the scalar curvatures of (M, h) ∂ ∂ u(y) = 0 and so ∂ρ (ρu) = u. By Since y is a critical point of u, one has ∂ρ −2 comparing this with (3.40), we get ρ (y) = s¯/6 − s¯N /8. Now if ρ(y) > ρ0 then take ρ1 = ρ0 and there is nothing to prove. On the other hand, if ρ(y) ≤ ρ0 , then (BC) gives an upper bound on s¯/6 − s¯N /8 which implies a lower bound on ρ(y). 17

In the following discussion we re-scale u such that m = min u = 1. Notice that Prop. 8 and Cor. 9 are valid for any m > 0. Conformally related metrics. At this point it’s useful to consider the conformally related metrics ˆ = uh . h

(3.41)

ˆ The Einstein equations on (N, g) induce the following equations on (M, h)  ˆ = 3 (d log u) ◦ (d log u) − 6h ,  Ric 2 (3.42)  ˆ −1 ∆ log u = 4u . ˆ are quasiFrom the estimate (3.2), we conclude that the two metrics h and h isometric on By (R), and so diamhˆ By (R) 6 R1 ,

(3.43)

where R1 depends only on R. By Lemma 7 there is a lower bound on the volume of the unit ball By (1) in (M, h). It follows volhˆ By (R) > volhˆ By (1) > volh By (1) > ν0 ,

(3.44)

for R > 1 and ν0 depending on distg¯(y, ∂N ). As to the Ricci curvature, (3.42) and Lemma 5 give ˆ 6 3 |∇u| ˆ 2 + 6|h| 6 λ , |Ric| 2u2

(3.45)

where λ depends only on R. In order to obtain an upper bound on the L2 1 2 ˆ we notice that |R| ˆ 2 = |W ˆ |2 + 1 |ˆ norm of the curvature of h, z |2 + 24 sˆ , where 2 ˆ zˆ and sˆ denote the trace-free Ricci and scalar curvature of h. Now (3.42) and Lemma 5 imply Z 1 1 2 z | + sˆ2 )dvolhˆ 6 C · volhˆ By (R) , (3.46) ( |ˆ 24 By (R) 2 and Lemma 6 implies volhˆ By (R) 6 V , where C and V depend only on R. It follows Z Z Z 1 2 2 2 ˆ |2 dvolˆ 6 Cˆ , (3.47) ˆ |W |R| dvolhˆ 6 (|ˆ z | + sˆ )dvolhˆ + h 24 M By (R) By (R) 18

where Cˆ = C · V + λ0 depends on R and ω0 . We are ready to state the main result of this section. Theorem 11. Let gi = hi + u2i dθ2 ∈ ESm,α 1 ,AH be a sequence of metrics on 5 4 1 N = M × S that satisfy the Boundary Condition, with ui > 0. In addition assume that the inclusion map i : ∂M → M induces a surjection ¯ , R) → 0 , H2 (∂M, R) → H2 (M

(3.48)

and that there is a positive constant ω0 < ∞ such that W idg¯i 6 ω0 .

(3.49)

Let yi ∈ M be a point where the minimum of ui occurs. Then there exists a subsequence of (M, hi , yi ) which converges to a complete metric (L, h, y∞ ) with y∞ = lim yi . The convergence is in the C ∞ topology, uniformly on compact subsets, and the manifold L weakly embeds in M, L ⊂⊂ M ,

(3.50)

i.e. smooth bounded domains in L embed as such in M. Moreover the manifold (L × S 1 , h + u2 dθ2 ) is Einstein, where u is defined by u(lim xi ) = lim ui (xi )/ui (yi ) for any sequence xi ∈ M . Proof. Re-scale ui so that ui (yi ) = 1. First we show that, after this re-scaling, the Boundary Condition stays valid (for a possibly smaller value of ρ0 ). It’s enough to show that ui (yi ) stays away from 0. Take any x ∈ S(ρ0 /2), where ρ0 is provided by the Boundary Condition. Then Lemma 5 and the upper bound on the width of (N, g¯i ) imply that mi = ui (yi ) > β0 ui (x) ,

(3.51)

where β0 is independent of i. On the other hand, the Boundary Condition gives a lower bound on ρi ui in the ρ0 -distance of the boundary. It follows that there exists a constant c > 0 such that (ρ0 /2)ui (x) > c. This and the above inequality imply that there is a uniform lower bound on mi . We divide the rest of the proof into four steps. Step 1. In this step, we show that for each R > 0, there exists a subsequence of (Byi (R), hˆi , yi ) that converges, in the C 1,α topology, to a C 1,α 19

ˆ y∞ ) with yi → y∞ . Recall that yi is chosen such Riemannian manifold (V, h, that ui (yi ) = min ui . The bounds (3.39) and (3.49) give ρ1 6 distg¯i (yi , ∂N ) 6 ω0 .

(3.52)

The Boundary Condition implies that the bounds (3.14) and (3.15) are valid for some τ¯ and T . Then (3.52) and Lemma 7 imply that (3.17) is valid on ByMi (R) and hence (3.44) follows. Now consider the sequence (Byi (R), hˆi , yi ) and observe that (3.43)–(3.47) imply that a subsequence (denoted again by hˆi ) is converging, in the Gromov-Hausdorff topology, to a 4-dimensional orbˆ y∞ ) which is a C 0 Riemannian metric and C 1,α off the singular ifold (V, h, points. The topological assumption (3.48) is made exactly to rule out orbifold degenerations (cf. [2]) and so we conclude that a subsequence of the sequence (Byi (R), hˆi , yi ) converges in the C 1,α topology to a C 1,α Riemannian ˆ y∞ ) with yi → y∞ . metric (V, h, Step 2. Now we proceed to control u further in order to control the Ricci curvature of (M, h). By Lemma 5, both u and |du| are bounded in Byi (R) and so one has a C 1,α bound for v = log u. Step 1 implies that there is a uniform lower bound h0 on the C 1,α harmonic radius of (Byi (R), hˆi ). ˆ From For x ∈ Byi (R) take the ball of radius h0 centered at x in (M, h). ˆ = 4 exp(−v) this, Schauder estimates, applied to the elliptic equation ∆v 2,α (cf. (3.42)), give a C bound on v and hence on u on a smaller ball (cf. [11, Theorem 6.2]) i.e. ∀x ∈ Bx (r0 /2) : |v|2,α 6 c ,

(3.53)

where c is independent of i, although it depends on r0 and R. By (0.5), this gives a uniform C 0 bound on the Ricci curvature of hi on Byi (R) |Richi | 6 λ ,

(3.54)

for some λ > 0 independent of i. Step 3. In this step we use standard elliptic regularity results to draw the promised C ∞ convergence. Prop. 8 and Cor. 9 imply Z |Rhi |2 dVhi 6 Λ , (3.55) Byi (R)

20

where Λ is independent of i. Now Lemma 7, (3.54) and (3.55) imply that a subsequence of (Byi (R), hi , yi ) converges in the C 1,α topology to a C 1,α Riemannian metric (By∞ (R), h, y∞ ). It follows that (Byi (R) × S 1 , gi , yi ) converge to (By∞ (R) × S 1 , g, y∞ ), in the C 1,α topology, where the metric on Byi (R) × S 1 is given by gi = hi + u2i dθ2 . Since these metrics are Einstein, the C 1,α convergence implies C ∞ convergence. Now take R → ∞ and the pointwise convergence of (M, hi , yi ) follows. Step 4. Finally we prove that the function u(lim xi ) = lim ui (xi ) is well defined and equations (0.5) are valid on (L, h). For the sequence {yi }, we have u(y∞ ) = lim ui (yi ) = 1. For any sequence xi → x∞ , we have disthi (xi , yi ) 6 T , for some T > 0. The discussion in Step 3 implies that u and its derivatives are bounded on Byi (T ) and so {ui } is an equicontinuous family of functions on compact sets. It follows that lim ui (xi ) is well-defined. As soon as u is defined as a weak limit of ui , the C ∞ convergence obtained in Step 3 and taking limit from both sides of (0.5) imply the last assertion in the theorem. Static circle actions with fixed points. So far we have assumed u > 0. We can remove this assumption as follows. Define T = {x ∈ N | u(x) = 0} , (3.56) and let M = (N − T )/S 1 be the quotient manifold. M is a 4-manifold with boundary T . The equations (0.5) are still valid on M . In the following discussion, let B(ρ0 , ρ) = B(ρ)\B(ρ0 ). Proposition 12. Under the same conditions as in Theorem 11, this time with the possibility of u = 0, the same conclusion holds for a subsequence of (Bi (ρ0 , ρ), hi , xi ), where xi satisfy (3.16), with the difference that L is a manifold with boundary ∂L. Proof. In the sequel we drop the subscript i. The Boundary Condition implies that, for all x with ρ(x) < ρ0 c < ρu < d ,

(3.57)

where c, d0 > 0 are constants. Without loss of generality, we can assume c < ρ0 . Since u is subharmonic, the maximum of u on the set B(ρ0 ) is 21

realized by some x0 with ρ(x0 ) = ρ0 . The inequality (3.57) implies that, for all x, u(x) 6 d/ρ0 + d/ρ(x) , (3.58) and for x with ρ(x) 6 ρ0 , u(x) > c/ρ(x) > c/ρ0 .

(3.59)

Now we show that there exists a constant a > 0 such that for all x ∈ B(c/2) one has Bx (a) ⊂ B(c) . (3.60) Take η(t) to be a minimizing geodesic of speed one from x to z ∈ S(c). To prove (3.60), it is enough to show that the length of this curve in (M, h) has a lower bound. We can assume that z = η(t) is the first point of the intersection of η with S(c). One has Z t Z t 1 1 t= |η| ˙g= |η| ˙ g¯ > L(η) , (3.61) c 0 0 ρ ¯ Now (BC) provides where L(η) denotes the length of the curve η(t) in (M, h). a lower bound for L(η) and (3.60) follows. Equations (3.59) and (3.60) imply u > 1 on Bx (a) for all x ∈ B(c/2). As in Lemma 5 we define f (x) = log u(x) > 0 and use [8, Theorem 6] to conclude that for all x with ρ(x) 6 c/2 |du| 6 G0 (ρ) ,

(3.62)

where G0 depends on ρ through (3.59). From this, the same proof as in Prop. 8 gives Z u|z|2 dVg 6 K 0 (ρ) .

(3.63)

B(ρ)

This and (3.59) imply that, for ρ < ρ0 Z |z|2 6 K(ρ) .

(3.64)

B(ρ0 ,ρ)

In this case, the bound on the Weyl curvature is automatic Z Z 2 |Wh | dVh = |Wh¯ |2 dVh¯ 6 λ00 (ρ) . B(ρ0 ,ρ)

(3.65)

B(ρ0 ,ρ)

The above two inequalities give the bound on the L2 norm of the curvature on B(ρ0 , ρ). From here on the same proof as in Theorem 11 works. 22

4

The Boundary Behavior

Assume g is an AHE metric on N n+1 with boundary metric γ and associated geodesic defining function ρ. Let A denote the second fundamental form of the level sets S(ρ) of ρ and H = trA. One has the following system of equations, cf. [16]. ρ¨ gρ − ng˙ ρ − 2Hgρ = ρ{2Ricρ − H g˙ ρ + (g˙ ρ )2 } , ρH˙ − H = ρ|A|2 , δA = −dH ,

(4.1) (4.2) (4.3)

where gρ is induced by g¯ = ρ2 g on S(ρ), Ricρ is its Ricci curvature and dot denotes the differentiation w.r.t. ρ. In case n = 4, the formal solution to this system, given by the Fefferman-Graham expansion, is the following (¯ g )ρ ∼ g(0) + ρ2 g(2) + ρ4 (log ρ)h(4) + ρ4 g(4) + . . . ,

(4.4)

where g(0) = γ is the boundary metric. This formal series does not need to converge in general. However, if the boundary metric is analytic, by results in [16], (4.1) admits formal solutions involving log ρ, the series converge and solutions are holomorphic functions of ρ and ρ ln ρ, when ρ is small. These results give solutions to Einstein equations in a collar neighborhood of the boundary, given the analytic boundary metric g(0) and the trace-free part of g(4) . Fuchsian systems and stability of solutions. Let u = (u1 , ..., um ) be a function of (x1 , ..., xn ) ∈ Ω ⊂ Cn . Assume f1 , ..., fl are holomorphic functions in a neighborhood of Ω × {0}1+m+mn . Let A(x) be a matrix with arrays holomorphic in x. Moreover let t0 , ..., tl be the time variables which we assume are close to zero in C. Then a Fuchsian PDE is a partial differential equation of the type (N + A)u =

l X

tp fp (x, t0 , ..., tp , u, ux ) ,

(4.5)

p=0

P where N = i,j nij ti ∂/∂tj . Regarding such systems the following existence result holds:

23

Theorem Pl 13. (Kichenassamy [15]) Consider the Fuchsian system (4.5) where N = k=0 (tk + ktk−1 )∂/∂tk . Suppose f is analytic near (0, 0, 0, 0) without constant terms in t0 = (t0 , ..., tl ). Moreover assume that A is a constant matrix with no eigenvalues with negative real part. Then (4.5) has near the origin exactly one analytic solution which vanishes for t0 = 0. The proof is rather technical; we use the proof of this theorem in order to establish Prop. 14 below. Fuchsian reduction. From now on we focus on the case n = 4, although the discussion is correct in all dimensions. Therefor let g be an AHE metric on N 5 with geodesic defining function r and analytic boundary metric g(0) = γN . The formal solutions to the system above exist with coefficients being holomorphic functions of r and r ln r. Following [16] one writes g¯ = ρ2 g = g(0) + r2 p ,

(4.6)

where p is given by p=q+

7 X

r7 (ln r)k uk (r, r ln r) ,

(4.7)

k=0

for some q that contains all terms of lower total degree in r (including terms up to the fourth order). Let ~u = (u0 , ..., u7 ); each ui is a 2-tensor on ∂N . Decomposing ~u = u~1 + u~2 to the sum of traceless part and a part proportional to g(0) , we have the following first-order Fuchsian system for an unknown ~v (defined below), (N + B)~v = r0 φ(x, r0 , r1 , ~v , ∂~v ) , (4.8) where r0 and r1 correspond to r and r ln r respectively, and N = r0 ∂/∂r0 + r1 ∂/∂r1 is a differential operator on the algebra B1 of formal series in r0 and r1 (with analytic coefficients), B is a matrix and ~v = (vK )16K612 = (~u1 , N~u1 , (r0 ∂i~u1 )16i64 , ~u2 , N~u2 , (r0 ∂i~u2 )16i64 ) .

(4.9)

Following a suggestion by Kichenassamy [17], in the following proposition, we prove a stability result.

24

i i Proposition 14. Suppose g(0) and g(4) are analytic on ∂N and i i (g(0) , g(4) ) → (g(0) , g(4) ) ,

(4.10)

with convergence in the C 2 topology on the boundary to analytic tensors g(0) , g(4) . Then the associated solutions g¯i converge to a solution g¯ as holomorphic functions of x, r and r ln r (polyhomogeneous convergence). Moreover they converge in the C ∞ topology of metrics in (0, ) × ∂N , on compact sets, for some  > 0. Proof. One needs to verify the polyhomogeneous convergence and the second part of the theorem follows. Thus we need to show that if (g(0) , g(4) ) and 0 0 ) are close in the C 2 topology on ∂N , then g¯ and g¯0 are close in the (g(0) , g(4) C ∞ topology in a collar neighborhood of ∂N in N . The Fuchsian equation 0 0 (4.8) for (g(0) , g(4) ) becomes (N + B)w ~ = r0 φ(x, r0 , r1 , w, ~ ∂ w) ~ ,

(4.11)

0 0 where w ~ is defined in terms of g(0) , g(4) . Define: ~y = ~v − w. ~ Comparing (4.8) and (4.11), we have

(N + B)~y = r0 F0 (x, r0 , r1 , ~v , w, ~ ∂~v , ∂ w) ~ = r0 F0 [~y ] .

(4.12)

Now we apply the proof of [15, Theorem 2.2] to finish the proof. We briefly sketch the ideas of the proof of [15, Theorem 2.2]. The Fuchsian system (4.5) with two time variables (T, Y ) = (r0 , r1 ) becomes: (N + A)z = f (T, Y, X, z, Dz) , z(0, 0, X) = z0 (X) ∈ ker(A) ,

(4.13)

where D = DX and f = T f0 = 0 for T = Y = 0. Notice that T = Y = 0 corresponds to the boundary and z corresponds to ~y in our case, while z0 corresponds to the boundary data which is solely a function of X (a parameter representing the boundary coordinates). Our goal is to control z in terms of z0 and constants that depend only on the boundary metric. The proof of [15, Theorem 2.2] is in five steps that we sketch below. Define F [z] = f (T, Y, X, z, Dz) .

25

(4.14)

Step 1. First one shows that the system (N + A)z(T, Y ) = k(T, Y ) , z(0, 0) = 0 ,

(4.15)

where k is analytic and independent of X and vanishes for T = Y = 0, has a unique analytic solution, given by Z 1 z(T, Y ) = H[k] = σ A−I k(σT, σ(T ln σ + Y ))dσ . (4.16) 0

Therefore rewrite the equation (4.13) as z = H[F [z]]. Let u = F [z] and consider u = G[u] = F [H[u]] . (4.17) One proceeds to solve this equation by a fixed point argument. Step 2. One defines two norms as follows. Assume f is analytic for X ∈ C n and d(X, Ω) < 2s0 and u ∈ C m with |u| < 2R, for some s0 , R > 0, where Ω is a bounded open neighborhood of 0. For a function u = u(X) define the s-norm kuks = sup{|u(X)| : d(X, Ω) < s} .

(4.18)

For a function u = u(T, Y, X), and α > 0 sufficiently small, define the α-norm s ( ) δ 0 |u|α = sup δ0−1 kuks (T, Y )(s0 − s) 1 − , α(s 0 − s) δ0 (T,Y )<α(s0 −s), 06s
(4.20)

where C1 is a constant. Step 4. Moreover, for 0 < s0 < s < s0 , if kuks , kvks 6 R, one has kF [u] − F [v]ks0 (T, Y ) 6 26

C2 δ0 (T, Y ) ku − vks s − s0

(4.21)

Step 5. Now assume that |u|α , |v|α < R/(2C2 α) and let G[u] = F [H[u]]. One proves the following inequality |G[u] − G[v]|α 6 C3 α|u − v|α ,

(4.22)

for some constant C3 . Now define u0 = 0 and u1 = G[u0 ] and choose R0 so that ku1 ks0 6 R0 δ0 (T, Y ) , (4.23) if |T | + θ|Y | = δ0 . Now choose α so small that C3 α < 1/2 and R0 s0 < R/4C1 α .

(4.24)

It follows that G is a contraction in the α-norm on {|u|α 6 R/(4C1 α)} and so by contraction mapping principle, a unique solution exists in this set. End of proof of Prop. 14. Since y is a solution inside the set above, we have |y|a 6 R0 s0 . (4.25) 0 0 Since the two initial pairs (g(0) , g(4) ) and (g(0) , g(4) ) are uniformly close in the 2 C topology on ∂N , u1 = G[0] = F [0] is small and so correspondingly one chooses R0 small. It follows from the above inequality and definitions in Step 2 of the proof above that

kyks < C(s, s0 , δ0 , R0 , a) ,

(4.26)

where C(s0 , δ0 , R0 , a) = o(R0 ). This implies that the uk terms (and in fact their first derivatives) in the expressions of g¯ and g¯0 are uniformly close. Since q involves only a finite number of terms and, by assumption, the first few terms and their derivatives are uniformly close, we conclude that g¯ and g¯0 are close in the C 0 topology. In another words: 0 0 kg(0) − g(0) kC 2 + kg(4) − g(4) kC 2 <  ⇒ k¯ g − g¯0 kC 0 < δ() ,

(4.27)

and the first part of the theorem follows. The second part of the theorem follows from the first part, since ln r is a C ∞ function with bounded derivatives on any compact set away from the boundary, cf. (4.6) and (4.7). Notice that C 0 closeness of these metrics implies their C ∞ closeness since these are solutions to elliptic PDE away from the boundary. 27

Corollary 15. Under the same assumptions as in the Prop. 14, we have ¯ for β ∈ (0, 1). g¯i → g¯ in the C 3,β topology on N Proof. The log terms in the F-G expansion in dimension 5 do not appear until the 4th order, and so the polyhomogeneous convergence obtained in Prop. 14 implies the C 3,β convergence for any β ∈ (0, 1). The C 3,β convergence above cannot be strengthened to C 4 or higher, in particular the usual C m,α harmonic radius near the boundary, when m > 3, may not be bounded by the boundary data. Our goal is to obtain the C 3,β control of the compactified metric near the boundary from a bound on the analytic boundary metric in the C m topology, for some m. Then the Boundary Condition (BC) (that is a weaker C 2,α control of the compactified metric) will follow. We first define the notion of logarithmic C 4,α harmonic radius. The logarithmic C 4,α harmonic radius. Definition 16. Let ρ be a geodesic defining function for an AHE metric g on N 5 and define g¯ = ρ2 g. Let Nρ = {x ∈ N | ρ(x) > ρ} , N ρ = {x ∈ N | ρ(x) 6 ρ} .

(4.28)

Let γ, g(2) , . . . be as in the F-G expansion in (2.15). Fix ρ0 , C > 0 and define a good r-covering of N to be a covering Us , s ∈ J = J1 ∪ J2 satisfying the following conditions: • (r1) δ =

1 −C e r 10

is a Lebesgue number for the covering {Us }.

• (r2) ∀s ∈ J1 – (r2-1) There are charts φs : B 4 (r) × [0, r) → Us ⊂ N such that the coordinate along [0, r) is just ρ and other coordinates xi are constant along integral curves of ρ. – (r2-2) φs (B 4 (r)) ⊂ ∂N and φs (B 4 (r/10)), s ∈ J1 , cover ∂N . – (r2-3) φ−1 : Us ∩ ∂N → B 4 (r) are harmonic coordinates on the s boundary.

28

– (r2-4) For all j = 0, . . . , 6 and γ = (γ1 , γ2 ) with |γ| = j, |γ1 | 6 4 and ∂ γ = ∂ργ1 ∂xγ2 : sup rj+α k∂ γ f kα 6 C ,

(4.29)

Us

where f = g¯ρ − γ − ρ2 g(2) − (ρ4 log ρ)h(4) . • (r3) ∀s ∈ J2 – (r3-1) Us ⊂ Nr/10 . – (r3-2) There are charts φs : B 5 (r) → Us ⊂ N such that φ−1 : s Us → B 5 (r) are harmonic coordinates. – (r3-3) For all j = 0, . . . , 4 and γ with |γ| = j sup rj+α k∂ γ g¯kα 6 C .

(4.30)

Us

C C • (r4) ∀s ∈ J : |dφ−1 s | 6 e and |dφs | 6 e on their domain of definition.

Finally define the logarithmic C 4,α harmonic radius, r(N ; g¯) to be the supremum of all r such that a good r-covering exists. By Prop. 14, upper bounds on the C 2 norms of g(0) and g(4) give further control of the metric. In particular, Prop. 14 implies that the AHE metric g satisfies the strong control property [2, Def. 4.2] i.e. the control of the metric in a weak norm implies control of the metric in a stronger norm. In the sequel, we illustrate how the work in [2], more specifically Th. 4.4 through Cor. 4.10, is applicable to our setting. Then it follows from [2, Cor. 4.10] (which holds in all dimensions where the strong control property holds) that the C 3,β geometry of the geodesic compactification g¯ is uniformly controlled in a neighborhood of the boundary. Our goal is to replace the usual harmonic radius with the logarithmic harmonic radius and show that the proofs of Th. 4.4 trough Cor. 4.10 of [2] stay valid. We do this by making the following two observations and refer the reader to the before-mentioned reference for more details. i) The logarithmic harmonic radius has the same scaling properties of the harmonic radius. ii) The logarithmic harmonic radius is lower semi continuous in the polyhomogeneous topology. To see this, suppose the F-G expansions of g¯i converge to g¯0 as analytic functions of x, ρ and log ρ. We show that ri > r − , 29

for i large, where r, ri are the logarithmic harmonic radii of g¯ and g¯i . Recall the definition of the logarithmic C 4,α harmonic radius. Since the polyhomogeneous convergence of gi to g implies at least the C 3,α convergence, we see that the conditions r1, r2-1, r2-2, r3-1 and r4 are all valid on (N, g¯i ) for a slightly smaller choice of r. Also since the boundary metrics ρ2i gi |∂N converge to ρ2 g|∂N in the C 2 topology on the boundary, both conditions r2-3 and r3-2 are valid (again for a possibly smaller r) along a subsequence. Finally both r2-4 and r3-3 follow from Prop. 14, since the convergence is in the C ∞ topology in a collar neighborhood of the boundary as functions of x, ρ and ρ log ρ. This proves the semi-lower continuity of the logarithmic harmonic radius. It follows from the discussion above and [2, Cor. 4.10] that: 4,α Corollary 17. Let N = M × S 1 be as before. Suppose g ∈ EAH has an analytic boundary metric γ satisfying kγkC 4,α ≤ K in a fixed coordinate system on ∂N . Then there exists ρ0 = ρ0 (K) > 0 such that the C 3,α geometry of the geodesic compactification g¯ is uniformly controlled in N ρ0 .

Now we are in the position to prove the following proposition. The proof is immediate from Theorem 11, Prop. 8 and Cor. 17. Proposition 18. Let gi be a sequence of strictly globally static AH Einstein metrics on N = M × S 1 , with strictly globally static and analytic boundary metrics γi on ∂N = ∂M ×S 1 and suppose γi → γ in the C 4,α topology on ∂N . ¯ induces a surjection Moreover suppose that the inclusion map i : ∂M → M ¯ , R) → 0 , H2 (∂M, R) → H2 (M

(4.31)

and that there is a constant ω0 < ∞ such that W idg¯i N 6 ω0 .

(4.32)

Then a subsequence of gi converges smoothly and uniformly on compact subsets to an AH Einstein metric g on N with boundary metric γ. In a collar neighborhood of ∂N , the geodesic compactifications g¯i converge in the polyhomogeneous topology to the geodesic compactification g¯ of g. This convergence is in the C ∞ topology away from the boundary. Now we present the proofs of Theorems A and B. Proof of Theorem A. To prove that Π0 is proper, we need to show 30

that for any sequence of analytic boundary metrics γi → γ ∈ Cω0 and AH Einstein metrics gi ∈ ES 1 ,AH with Π0 ([gi ]) = [γi ], there is a subsequence of gi converging to an AHE metric g. We consider the following two cases: Case i. There is a constant s0 > 0 such that for the scalar curvature sγi of γi sγ0 > s0 . (4.33) In this case one has: s0g¯i =

¯ 2 ρ|2 ¯ 2 8|D 2(∆ρ) 1 > = ρs2g¯i . ρ ρ 32

(4.34)

This gives (1/sg¯i )0 6 ρ/32. Integrating this√ inequality implies that the length √ of each minimizing ρ-geodesic is at most 4 3/ s0 i.e. √ 4 3 W idg¯i 6 √ . (4.35) s0 And so the theorem follows from Prop. 18. Case ii. The only case remaining is when sγi > 0 and W idg¯i → ∞ .

(4.36)

In this case, we use the rigidity result of [1, Remark 5.2, Lemma 5.5]. Very briefly, the sequence (N, gi , yi ) converges in the Gromov-Hausdorff topology to a hyperbolic cusp metric with flat boundary metric. This contradicts the condition γ ∈ C 0 and the proof is completed.  0 At this point, we consider connected components of E1,AH and restrict Π0 to these components. Before we present the proof of Theorem B, we need to show that the definition (0.9) of deg Π0 is well-defined in the context of anaω lytic metrics. Notice that we have proved Π0 is proper on E1,AH but it’s still an open problem whether this space has a manifold structure. On the other hand, Prop. 4 gives a manifold structure for the space E˜1,AH . By Sard-Smale theorem [20], regular values of the Fredholm map Π are open and dense in C ∞ . Since C ω is dense in C ∞ , there exist many analytic metrics among the regular values of Π. It follows that the definition (0.9) of deg Π0 is meaningful for such analytic regular values of Π and [2, Theorem 5.1] shows that this

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definition is independent of the choice of these analytic regular values. Now we are ready to present the proof of Theorem B. Proof of Theorem B. As the seed metric we take g0 , the metric induced from the hyperbolic metric on B5 /Z, where the Z action is given by a fixed translation along a geodesic. The conformal boundary of g0 is the standard product metric on S 3 × S 1 (log L). The Isometry Extension Theorem [4, Th. 2.4] implies that any AH Einstein metric with this conformal infinity has SO(3) × SU (2) acting effectively by isometries. The only such metrics are the hyperbolic metric on B5 and the Ads-Schwarzschild metric on S 3 × R2 . The latter is defined on a different manifold than B5 and so (Π0 )−1 [γ] = [gH ]. It follows deg Π0 = 1 . (4.37) In particular, Π0 is onto the component containing the standard product metric. 

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[7] O. Biquard, Metriques d’Einstein asymptotiquement symmetriques, Asterisque, 265 (2000). [8] S. Y. Cheng and S. T. Yau, Differential Equations on Riemannian Manifolds and their Geometric Applications, Comm. Pure Appl. Math., 28 (1975) 333-354. [9] P. Chru´sciel, E. Delay, J.M. Lee and D. Skinner, Boundary regularity of conformally compact Einstein metrics, J. Differential Geometry 69 (2005) 111-136. [10] C. Fefferman, and C.R. Graham, Conformal invariants, Elie Cartan et les Mathematiques d’aujourd’hui, Asterisque, hors serie, Soc. Math. France, Paris (1985) 95-116. [11] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, second edition, Springer-Verlag (1983). [12] C.R. Graham, Volume and area renormalization for Conformally compact Einstein metrics, Preprint, Sep 1999, math.DG/9909042. [13] C.R. Graham and J.M. Lee, Einsten metrics with prescribed conformal infinity on the ball, Adv. in Math., vol. 87 (1991) 186-225. [14] S.W. Hawking and D.N. Page, Thermodynamics of black holes in Antide Sitter space, Comm. Math. Phys., 87 (1983) 577–588. [15] S. Kichenassamy, Nonlinear Wave Equations, Marcel Dekker, Inc., New York, 1996. [16] S. Kichenassamy, On a conjecture of Fefferman and Graham, Advances in Math. 184 (2004) 268-288. [17] S. Kichenassamy, Private Communication, Fall 2004. [18] J.M. Lee, Fredholm Operators and Einstein metrics on conformally compact manifolds, Memoirs of the American Mathematical Society, 183 (2006). [19] P. Petersen, Riemannian Geometry, Springer-Verlag, New York Inc., 1998. 33

[20] S. Smale, An infinite dimensional version of Sards Theorem, Amer. J. Math. 87 (1965) 861-866. [21] E. Witten, Anti De Sitter spaces and holography, Adv. Theor. Math. Phys., 2 (1998) 253–291. [22] S.T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure & Appl. Math., vol. 28 (1975) 201-228.

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