BOUNDARY REGULARITY OF CONFORMALLY COMPACT EINSTEIN METRICS

PIOTR T. CHRUŚCIEL, ERWANN DELAY, JOHN M. LEE, AND DALE N. SKINNER

Abstract. We show that C^{2} conformally compact Riemannian Einstein metrics have conformal compactifications that are smooth up to the boundary in dimension 3 and all even dimensions, and polyhomogeneous in odd dimensions greater than 3.

1. Introduction

Suppose \overline{M} is a smooth, compact manifold with boundary, and let M denote its interior and ∂M its boundary. (By “smooth,” we always mean C^{∞}.) A Riemannian metric g on M is said to be conformally compact if for some smooth defining function ρ for ∂M in \overline{M}, ρ^{2}g extends by continuity to a Riemannian metric (of class at least C^{0}) on \overline{M}. The rescaled metric \overline{g} = ρ^{2}g is called a conformal compactification of g. If for some (hence any) smooth defining function ρ, \overline{g} is in C^{k}(\overline{M}) or C^{k,λ}(\overline{M}), then we say g is conformally compact of class C^{k} or C^{k,λ}, respectively.

If g is conformally compact, the restriction of \overline{g} = ρ^{2}g to ∂M is a Riemannian metric on ∂M, whose conformal class is determined by g, independently of the choice of defining function ρ. This conformal class is called the conformal infinity of g.

Several important existence and uniqueness results [1,2,5,9,13] concerning conformally compact Riemannian Einstein metrics have been established recently. For many applications to physics and geometry, it turns out to be of great importance to understand the asymptotic behaviour of the resulting metrics near the boundary. This question has been addressed by Michael Anderson [2], who proved that if g is a 4-dimensional conformally compact Einstein metric with smooth conformal infinity, then the conformal compactification of g is smooth up to the boundary in suitable coordinates. It has long been conjectured that in higher dimensions, conformally compact Einstein metrics with smooth conformal infinities should have infinite-order asymptotic expansions in terms of ρ and log ρ. The purpose of this paper is to confirm that conjecture.

Date: January 27, 2004.

First author partially supported by Polish Research Council grant KBN 2 P03B 073 24; second author partially supported by the ACI program of the French Ministry of Research.

The choice of special coordinates in Anderson’s result cannot be dispensed with. Because the Einstein equation is invariant under diffeomorphisms, we cannot expect that the conformal compactification of an arbitrary conformally compact Einstein metric will necessarily have optimal regularity for all C^{∞} structures on \overline{M}. For example, suppose g is an Einstein metric on M with a smooth conformal compactification, and let Ψ: \overline{M} → \overline{M} be a homeomorphism that restricts to the identity map of ∂M and to a diffeomorphism from M to itself. Then Ψ^{∗}g will still be Einstein with the same conformal infinity, but its conformal compactification ρ^{2}Ψ^{∗}g may no longer be smooth. Thus the best one might hope for is that an arbitrary conformally compact Einstein metric can be made smoothly conformally compact after pulling back by an appropriate diffeomorphism. Even this is not true in general, because Fefferman and Graham showed in [7] that there is an obstruction to smoothness in odd dimensions.

Since Einstein metrics are always smooth (in fact, real-analytic) in suitable coordinates in the interior, only regularity at the boundary is at issue. For that reason, instead of assuming that \overline{M} is compact, we will assume only that it has a compact boundary component Y, and restrict our attention to a collar neighborhood of Y in \overline{M}, which we may assume without loss of generality is diffeomorphic to Y × [0, 1). Throughout this paper, then, Y will be an arbitrary smooth, connected, compact, n-dimensional manifold without boundary, and we make the following identifications:

\overline{M} = Y × [0, 1), M = Y × (0, 1), ∂M = Y × \{0\}.

Let ρ: \overline{M} → [0, 1) denote the projection onto the [0, 1) factor; it is a smooth defining function for ∂M in \overline{M}. For 0 < R < 1, we define

M_{R} = Y × (0, R], \overline{M}_{R} = Y × [0, R].

In this context, we extend the definition of conformally compact metrics by saying that a smooth Riemannian metric g on M or M_{R} is conformally compact if ρ^{2}g extends to a continuous metric on \overline{M} or \overline{M}_{R}, respectively. A continuous map Ψ: \overline{M}_{R} → \overline{M} (for some R) that restricts to the identity map of ∂M and to a (C^{∞}) diffeomorphism from M_{R} to its image will be called a collar diffeomorphism. If Ψ and its inverse are of class C^{k} (or C^{k,λ}) up to the boundary, we will call it a C^{k} (resp., C^{k,λ}) collar diffeomorphism.

The object of this paper is to prove that the following regularity holds:

Theorem A. Let g be a smooth Riemannian metric on M. Suppose that dim M = n + 1 ≥ 3; g is Einstein with Ric(g) = −ng; g is conformally compact of class C^{2}; and the representative γ = ρ^{2}g|_{∂M} of the conformal infinity of g is smooth. Let γ̃ be any smooth representative of the conformal class [γ]. Then for any 0 < λ < 1, there exists R > 0 and a C^{1,λ} collar diffeomorphism Φ: \overline{M}_{R} → \overline{M} such that Φ^{∗}g can be written in the form

(1.1)  Φ^{∗}g = ρ^{−2}(dρ^{2} + G(ρ)),

where {G(ρ) : 0 < ρ ≤ R} is a one-parameter family of smooth Riemannian metrics on Y, which has a continuous extension to \overline{M}_{R} with G(0) = γ̃, and has the following regularity:

(a) If dim M is even or equal to 3, then G extends smoothly to \overline{M}_{R}, so Φ^{∗}g is conformally compact of class C^{∞}.

(b) If dim M is odd and greater than 3, then G can be written in the

form

G(ρ) = φ(ρ, ρ^{n} log ρ),

with φ(ρ, z) a two-parameter family of Riemannian metrics on Y that is smooth in all of its arguments as a function on Y × [0, R] × [R^{n} log R, 0]. Furthermore, Φ^{∗}g is smoothly conformally compact if and only if ∂_{z}φ(0, 0) vanishes identically on ∂M.

Remark. The symmetric 2-tensor field ∂_{z}φ(0, 0) along ∂M can be determined in principle from local computations involving the conformal class [γ] (cf. [7,8]), and in fact the vanishing of this tensor field is a necessary condition for the existence of a smoothly conformally compact Einstein metric on M with [γ] as conformal infinity. Explicit formulae in low dimensions can be found in [10].

The main idea of the proof is to use the harmonic map equation to put g into a gauge in which it satisfies an elliptic equation, and then apply the polyhomogeneity results of [3]. The proof consists of four steps. First, we construct a preliminary collar diffeomorphism that makes ρ^{2}g coincide to second order along ∂M with a smooth product metric \overline{h}. Second, applying the inverse function theorem to the harmonic map equation, we show that there exists a collar diffeomorphism H: \overline{M}_{R} → \overline{M}_{R} that is harmonic in M_{R}, thought of as a map from (M_{R}, g) to (M_{R}, h), where h = ρ^{−2}\overline{h}. It follows that the metric g̃ = (H^{−1})^{∗}g satisfies the following “gauge-broken Einstein equation” near ∂M:

(1.2)  Q(g̃, h) := Ric(g̃) + ng̃ − δ∗ ; g̃(Δ_{g̃h}(Id)) = 0

(see, e.g., [13]), where Δ_{g̃h} is the harmonic map Laplacian. The third step is to show that solutions to (1.2) satisfy the hypotheses of [3, Theorem 5.1.1] and therefore are polyhomogeneous (i.e., have asymptotic expansions in powers of ρ and log ρ). The last step is to use a special defining function and Fermi coordinates near the boundary to put the metric into the form (1.1).

The first and third authors would like to acknowledge useful discussions with Michael Anderson and Robin Graham.

2. Weighted Hölder spaces

Throughout most of this paper, we will use the notations and conventions of [13]. We define M_{R} and \overline{M}_{R} as in the introduction. We assume throughout that dim M = n + 1 ≥ 3. Any smooth local coordinates θ = (θ^{1}, …, θ^{n}) on an open set U ⊂ Y yield smooth coordinates (θ, ρ) = (θ^{1}, …, θ^{n}, ρ) on the open subset Ω = U × [0, 1) ⊂ \overline{M}. Choose finitely many such charts (U_{i}) to cover Y, with each set U_{i} chosen so that the coordinate functions extend smoothly to a neighborhood of \overline{U}_{i} in Y. The resulting coordinates on Ω_{i} = U_{i} × [0, 1) ⊂ \overline{M} will be called background coordinates for \overline{M}.

Let B_{1}, B_{2} be fixed open coordinate balls in the upper half-space ℍ^{n+1} = {(x, y) = (x^{1}, …, x^{n}, y) : y > 0}, with (0, …, 0, 1) ∈ B_{1} ⊂ \overline{B}_{1} ⊂ B_{2}. We will use the summation convention, with Greek indices generally understood to run from 1 to n, and Roman indices to run from 1 to n + 1; sometimes it will be convenient to denote ρ by θ^{n+1} and y by x^{n+1}. Suppose p ∈ M_{R}, and let (θ_{0}, ρ_{0}) be the coordinate representation of p in some fixed background chart. If p is sufficiently close to ∂M, we can define a diffeomorphism Φ_{p}: \overline{B}_{2} → M by

(θ, ρ) = Φ_{p}(x, y) = (θ_{0} + ρ_{0}x, ρ_{0}y).

As is shown in [13], for R sufficiently small, there exists a countable set of points \{p_{i}\} ⊂ M_{R} such that the sets \{Φ_{p_{i}}(\overline{B}_{2})\} form a uniformly locally finite covering of M_{R}, and the sets {Φ_{p_{i}}(B_{1})} still cover M_{R}. For each such map, set Φ_{i} = Φ_{p_{i}}, V_{1}(p_{i}) = Φ_{i}(B_{1}) and V_{2}(p_{i}) = Φ_{i}(B_{2}). Then for each i, (V_{2}(p_{i}), Φ_{i}^{−1}) is a coordinate chart on M_{R}, called a Möbius chart; the corresponding coordinates (x, y) will be called Möbius coordinates. It is shown in [13, Lemma 2.1] that if g is any C^{k,λ} conformally compact metric on M, the pulled-back metrics Φ_{i}^{∗}g are all uniformly C^{k,λ} equivalent to the hyperbolic metric y^{−2} ∑_{j}(dx^{j})^{2} on \overline{B}_{2}. By compactness, they are also uniformly C^{k,λ} equivalent on \overline{B}_{2} to the Euclidean metric ∑_{j}(dx^{j})^{2}.

We will be working in weighted Hölder spaces whose norms reflect the intrinsic geometry of a conformally compact metric. Before introducing them, let us record some elementary facts about Hölder spaces on subsets of ℝ^{m}. If U ⊂ ℝ^{m} is a precompact open subset, k, p are nonnegative integers, and λ ∈ [0, 1), we denote by C^{k,λ}(\overline{U}; ℝ^{p}) the standard Hölder space of functions from \overline{U} to ℝ^{p}, and we denote the usual Hölder norm on this space by ∥·∥_{k,λ;\overline{U}}. If V ⊂ ℝ^{p} is any open set, C^{k,λ}(\overline{U}; V) denotes the open subset of C^{k,λ}(\overline{U}; ℝ^{p}) consisting of maps that take their values in V.

Lemma 2.1. Let U ⊂ ℝ^{m} and V ⊂ ℝ^{p} be convex, precompact open subsets, let k be a nonnegative integer, and let λ ∈ [0, 1).

(a) Each coordinate derivative defines a continuous linear map from C^{k+1,λ}(\overline{U}; ℝ^{p}) to C^{k,λ}(\overline{U}; ℝ^{p}).

(b) Pointwise multiplication defines a continuous bilinear map from C^{k,λ}(\overline{U}; ℝ) × C^{k,λ}(\overline{U}; ℝ) to C^{k,λ}(\overline{U}; ℝ).

(c) If u ∈ C^{k,λ}(\overline{U}; V) and f ∈ C^{k,λ}(\overline{V}; ℝ) with k ≥ 1, then f ∘ u ∈ C^{k,λ}(\overline{U}; ℝ), and there exists a constant C depending only on k, λ, U, and V such that

∥f ∘ u∥_{k,λ;\overline{U}} ≤ C∥f∥_{k,λ;\overline{V}} (1 + ∥u∥_{k,λ;\overline{U}}^{k+λ}) .

(d) If f ∈ C^{k+2}(V; ℝ) with k ≥ 1, then composition u ↦ f ∘ u defines a C^{1} map from C^{k,λ}(\overline{U}; V) to C^{k,λ}(\overline{U}; ℝ).

(e) Given f ∈ C^{k+2}(\overline{V}; ℝ) and u_{0} ∈ C^{k,λ}(\overline{U}; V) with k ≥ 1, there exists δ > 0 and a constant C = C(U, V, k, λ, f, u_{0}, δ) such that the following estimate holds for all u ∈ C^{k,λ}(\overline{U}; V) with ∥u − u_{0}∥_{k,λ;\overline{U}} ≤ δ:

∥f ∘ u − f ∘ u_{0}∥_{k,λ;\overline{U}} ≤ C∥u − u_{0}∥_{k,λ;\overline{U}}.

Proof. The first two assertions are elementary and are left to the reader, and (e) follows easily from (d). Proofs of (c) and (d) can be found in [14]. q.e.d.

Now we proceed to define our weighted Hölder spaces on M_{R}. Let E → M be any tensor bundle, and let R > 0 be chosen so that M_{R} is covered by Möbius charts as above. For an integer k ≥ 0 and λ ∈ (0, 1), we define the intrinsic Hölder space C^{k,λ}(M_{R}; E) to be the set of locally C^{k,λ} sections of E over M_{R} whose component functions in Möbius coordinates satisfy a uniform C^{k,λ} bound, with norm

∥u∥_{k,λ} := sup_{i} ∥Φ_{i}^{∗}u∥_{k,λ;\overline{B}_{2}},

where the supremum is over the countable collection of Möbius charts described above. Weighted versions of these Hölder spaces are defined by setting C_{δ}^{k,λ}(M_{R}; E) = ρ^{δ}C^{k,λ}(M_{R}; E), with the norm ∥u∥_{k,λ,δ} := ∥ρ^{−δ}u∥_{k,λ}. It is shown in [13, Lemma 3.5] that this norm is equivalent to

∥u∥_{k,λ,δ} ∼ sup_{i} ρ(p_{i})^{−δ}∥Φ_{i}^{∗}u∥_{k,λ;\overline{B}_{j}}

for either j = 1 or j = 2.

3. A preliminary normalization

Suppose that g satisfies the hypotheses of Theorem A. Let γ̃ be an arbitrary smooth representative of the conformal class [γ] on Y. Define a smooth product metric \overline{h} on \overline{M} = Y × [0, 1) by

\overline{h} = dρ^{2} + γ̃.

Let h = ρ^{−2}\overline{h}, which is smoothly conformally compact and has [γ̃] as conformal infinity.

The goal of this section is to show that we can modify g by a collar diffeomorphism so that it agrees with h to second order along ∂M. We begin with the simple observation that they can be made to agree to first order by rescaling ρ.

Lemma 3.1. Let g and h be as above. If R > 0 is sufficiently small, there exists a C^{∞} collar diffeomorphism G_{0} : \overline{M}_{R} → \overline{M} that satisfies ρ^{2}G_{0}^{∗}g = ρ^{2}h + O(ρ) in any background coordinates.

Proof. We can write γ̃ = fγ for some smooth positive function f: Y → ℝ. It is easy to check that the conclusion of the lemma is satisfied by G_{0}(x, ρ) = (x, ρ/f(x)), as long as R is chosen smaller than the infimum of f so that G_{0} maps \overline{M}_{R} into \overline{M}. q.e.d.

Next we show that we can make g agree with h to one higher order along ∂M. This lemma requires somewhat more work than might be expected because we are only assuming that g has a C^{2} conformal compactification, and we need to make this normalization without losing any smoothness.

Lemma 3.2. Let g and h be as above. For any sufficiently small R > 0, there exists a C^{3} collar diffeomorphism G: \overline{M}_{R} → \overline{M} that satisfies ρ^{2}G^{∗}g = ρ^{2}h + O(ρ^{2}) in any background coordinates.

Proof. After replacing g by G_{0}^{∗}g, where G_{0} is given by the preceding lemma, we may as well assume that ρ^{2}g|_{T ∂M} = γ̃. We will begin by showing that there exists a C^{3} defining function r satisfying |dr|_{r^{2}g}^{2} = 1 + O(ρ^{2}). Let \overline{g} = ρ^{2}g, which is a C^{2} Riemannian metric on \overline{M}_{R}. Because Ric(g) = −ng, it follows that |dρ|2 ; \overline{g} = 1 along ∂M (cf. [9, p. 192]). By Taylor’s theorem, therefore, there is a function b ∈ C^{1}(\overline{M}_{R}) such that

|dρ|2 ; \overline{g} = 1 + bρ.

By Corollary 3.3.2 of [3], there exists a real-valued function r ∈ C^{3}(\overline{M}_{R}) ∩ C^{∞}(M_{R}) such that

r|_{∂M} = 0,
∂_{ρ}r|_{∂M} = 1,
∂_{ρ}^{2}r|_{∂M} = −b|_{∂M}.

Then r is a C^{3} defining function for ∂M, which satisfies r = ρ − \frac{1}{2}bρ^{2} + O(ρ^{3}), dr = (1 − bρ)dρ + O(ρ^{2}). Therefore,

(3.1)  \begin{aligned}|dr|_{r^{2}g}^{2} & = |dr|_{(r/ρ)^{2}\overline{g}}^{2} \\ & = (r/ρ)^{−2}|dr|2 ; \overline{g} \\ & = (1 − \frac{1}{2}bρ)^{−2}(1 − bρ)^{2}|dρ|2 ; \overline{g} + O(ρ^{2}) \\ & = 1 + O(ρ^{2}). \end{aligned}

Let P denote the gradient of r with respect to the metric r^{2}g; thus P is a C^{2} vector field on \overline{M}_{R}. Because P ρ = ⟨dr, dρ⟩_{r^{2}g} = ⟨dr, dr + O(ρ)⟩_{r^{2}g} = 1 + O(ρ), we can write P = ∂/∂ρ + Q, where Q is a C^{2} vector field on \overline{M}_{R} that is tangent to ∂M. Choose a smooth embedding X = (X^{1}, …, X^{N}): Y ↪ ℝ^{N} into some Euclidean space, and denote by the same symbol the extension of each coordinate function X^{A} to \overline{M} = Y ×[0, 1), chosen to be constant along the [0, 1) factor. By [3, Cor. 3.3.2] again, for each A = 1, …, N, there is a C^{3} function X̃^{A}: \overline{M} → ℝ, smooth in M, satisfying

X̃^{A}|_{∂M} = X^{A}|_{∂M},
∂_{ρ}X̃^{A}|_{∂M} = −QX^{A}|_{∂M},
∂_{ρ}^{2}X̃^{A}|_{∂M} = −∂_{ρ}QX^{A}|_{∂M}.

It follows that P X̃^{A} = O(ρ^{2}). The C^{3} map X̃: \overline{M} → ℝ^{N} whose coordinate functions are (X̃^{1}, …, X̃^{N}) thus satisfies X̃_{∗}P = O(ρ^{2}).

We wish to use X̃ and r to construct a collar diffeomorphism of \overline{M}_{R}. However, X̃ might not map into X(Y). To correct this, let U ⊂ ℝ^{N} be a tubular neighborhood of X(Y), and let Π: U → X(Y) be a smooth retraction. Define a C^{3} map Z: \overline{M}_{R} → \overline{M} by

Z(x, ρ) = (X^{−1} ∘ Π ∘ X̃(x, ρ), r(x, ρ)).

The restriction of Z to ∂M is the identity, and for some small ε > 0, Z is an embedding of \overline{M}_{ε} into M.

Let p ∈ M_{R} be arbitrary, and let q = Z(p). Writing Z_{∗}g = (Z^{−1})^{∗}g, we conclude from (3.1) that

|dρ(q)|_{ρ^{2}Z_{∗}g}^{2} = ρ(q)^{−2}|dρ(q)|_{Z_{∗}g}^{2}
= ρ(Z(p))^{−2}|d(ρ ∘ Z)(p)|_{g}^{2}
= r(p)^{−2}|dr(p)|_{g}^{2}
= |dr(p)|_{r^{2}g}^{2} = 1 + O(ρ(p)^{2}) = 1 + O(ρ(q)^{2}).

To check the mixed tangential/normal components of ρ^{2}Z_{∗}g, let (θ^{α}, ρ) be any background coordinates on \overline{M}_{R}, and let Z^{α} = θ^{α} ∘ Z denote the tangential component functions of Z in these coordinates. Observe that dθ^{α}(Z_{∗}P) = dθ^{α}(X_{∗}^{−1} ∘ Π_{∗} ∘ X̃_{∗}P). Because the component functions of Π_{∗} (as a map from ℝ^{N} to itself) are uniformly bounded, as are those of X_{∗}^{−1} in background coordinates, it follows that the tangential components P Z^{α} of Z_{∗}P in background coordinates are O(ρ^{2}). Thus

⟨dρ(q), dθ^{α}(q)⟩_{ρ^{2}Z_{∗}g} = ρ(q)^{−2}⟨dρ(q), dθ^{α}(q)⟩_{Z_{∗}g}
= ρ(Z(p))^{−2}⟨d(ρ ∘ Z)(p), d(θ^{α} ∘ Z)(p)⟩_{g}
= ⟨dr(p), dZ^{α}(p)⟩_{r^{2}g}
= P_{p}Z^{α} = O(ρ(p)^{2}) = O(ρ(q))^{2}.

We define our collar diffeomorphism by G = Z^{−1}|_{M_{R_{0}}} for R_{0} sufficiently small, and let ĝ = ρ^{2}G^{∗}g = ρ^{2}Z_{∗}g. By construction, in any background coordinates,

(3.2)  |dρ|2 ; ĝ = 1 + O(ρ^{2})
(3.3)  ⟨dρ, dθ^{α}⟩ĝ = O(ρ^{2}).

Inverting the coordinate matrix of ĝ, therefore, we find that

ĝ = dρ^{2} + ĝ_{αβ}(θ, ρ)dθ^{α}dθ^{β} + O(ρ^{2})

for some functions ĝ_{αβ} that are C^{2} up to ∂M. Moreover, because the restriction of G to ∂M is the identity and G^{∗}ρ = ρ+O(ρ^{2}), ĝ_{αβ} = γ̃_{αβ} = \overline{h}_{αβ} at points of ∂M.

To conclude the proof, we will use the Einstein equation to show that ∂_{ρ}ĝ_{αβ} = 0 along ∂M, which implies ĝ = \overline{h} + O(ρ^{2}) as desired. In terms of ĝ, the Einstein equation for G^{∗}g translates to

−nρ^{−2}ĝ_{jk} = R̂_{jk} + (n − 1)ρ^{−1}ρ_{;jk} + ρ^{−1}ρ_{;l}^{l}ĝ_{jk} − nρ^{−2}ρ_{;l}ρ_{;}^{l}ĝ_{jk},

where the semicolons indicate covariant derivatives, all taken with respect to ĝ (cf. [12, p. 266]). Multiplying by ρ, using (3.2), and evaluating at ρ = 0, we obtain (n − 1)ρ_{;jk} + ρ_{;l}^{l}ĝ_{jk} = 0 along ∂M. Taking the trace with respect to ĝ, we find that ρ_{;l}^{l} = 0 and therefore ρ_{;jk} = 0 along ∂M. Expanding this equation in terms of the Christoffel symbols of ĝ in background coordinates, we conclude that ∂_{ρ}ĝ_{αβ} = 0 along ∂M as claimed. q.e.d.

4. The harmonic map normalization

In this section, we will show that g can be modified by a collar diffeomorphism so that it satisfies the elliptic equation (1.2) near the boundary. We seek a collar diffeomorphism that is harmonic from (M_{R}, g) to (M_{R}, h), where h is the smoothly conformally compact metric defined in the preceding section. In order to find one, we will parameterise the diffeomorphisms near the identity by small vector fields using the Riemannian exponential map of h.

For any small R > 0, let ∂_{R}M_{R} = Y × {R} denote the “inner boundary” of \overline{M}_{R}, and let C̊_{δ}^{k,λ}(M_{R}; T M) denote the set of vector fields in C_{δ}^{k,λ}(M_{R}; T M) that vanish on ∂_{R}M_{R}. If v ∈ C̊_{δ}^{k,λ}(M_{R}; T M), define a map H_{v}: M_{R} → M by

H_{v}(p) = exp_{p}(v(p)),

where exp denotes the Riemannian exponential map of h. Since conformally compact metrics are complete at infinity [15], H_{v} is well-defined as a map from M_{R} into M as long as both R and v are sufficiently small.

Let us call a map H: M_{R} → M_{R} admissible if for each Möbius chart, H maps \overline{V_{1}(p_{i})} into V_{2}(p_{i}). Because h is uniformly equivalent to the Euclidean metric in Möbius coordinates, for any admissible map H, the Riemannian distance d_{h}(p, H(p)) is uniformly bounded for p ∈ M_{R}. This implies that d\overline{h}(p, H(p)) → 0 uniformly as p → ∂M, which in turn implies that any admissible map has a continuous extension to ∂M that fixes ∂M pointwise.

Lemma 4.1. If δ ≥ 0 and v is sufficiently small in C̊_{δ}^{1,0}(M_{R}; T M), then H_{v} is an admissible map from M_{R} to itself.

Proof. Note that

(4.1)  d_{h}(p, H_{v}(p)) = d_{h}(p, exp_{p}(v(p))) ≤ |v(p)|_{h}.

Because d_{h} is uniformly equivalent to Euclidean distance in Möbius coordinates, it follows that H_{v} will be an admissible map if ∥v∥_{0,0,δ} is sufficiently small (actually, for fixed δ > 0, it suffices to choose R small enough), provided that H_{v} maps M_{R} to itself.

To establish that H_{v}(M_{R}) ⊂ M_{R}, assume that ∥v∥_{1,0,δ} is small enough that |∇v|_{h} ≤ \frac{1}{2} on M_{R}. Let p ∈ M_{R} be arbitrary, and let γ: [0, b] → M be a unit-speed distance-minimizing h-geodesic from ∂_{R}M_{R} to p. At any time t such that v(γ(t)) ≠ 0, |v(γ(t))|_{h} is a differentiable function of t and

\frac{d}{dt} |v(γ(t))|_{h} = \frac{2 ⟨∇_{γ′(t)}v, v(γ(t))⟩_{h}}{2|v(γ(t))|_{h}}
≤ |∇_{γ′(t)}v|_{h} ≤ \frac{1}{2}.

Setting f(t) = |v(γ(t))| − \frac{1}{2}t, we see that f is continuous on [0, b] and satisfies f(0) = 0 (recall that v = 0 on ∂_{R}M_{R}). At any t such that |v(γ(t))| ≠ 0, f is differentiable and satisfies f′(t) ≤ 0 by the computation above, so that f(t) ≤ 0 for all t ∈ [0, b]. In particular, taking t = b = d_{h}(p, ∂_{R}M_{R}), this implies that |v(p)|_{h} ≤ \frac{1}{2}d_{h}(p, ∂_{R}M_{R}) for all p, which together with (4.1) implies that H_{v} maps M_{R} to itself. q.e.d.

Let Σ^{2} denote the bundle of symmetric covariant 2-tensors over M. For any section w of Σ^{2}, write g_{w} = h + w. For any 0 < λ < 1, define a map

Θ: C̊_{1+λ}^{2,λ}(M_{R}; T M)×C_{1+λ}^{1,λ}(M_{R}; Σ^{2}) → C_{1+λ}^{0,λ}(M_{R}; T M)×C_{1+λ}^{1,λ}(M_{R}; Σ^{2})

by

Θ(v, w) = ((H_{v})_{∗}^{−1}(Δ_{g_{w}h}H_{v}), w) ,

where Δ_{g_{w}h}H_{v} denotes the harmonic map Laplacian of H_{v}, viewed as a map from (M_{R}, g_{w}) to (M_{R}, h).

Lemma 4.2. The map Θ is well-defined and of class C^{1} in a neighborhood of (0, 0) in C̊_{1+λ}^{2,λ}(M_{R}; T M) × C_{1+λ}^{1,λ}(M_{R}; Σ^{2}).

Proof. When v and w are understood, let us write g = g_{w} and H = H_{v} for brevity. Recall that g is uniformly C^{1,λ} equivalent to the Euclidean metric in Möbius coordinates; a similar statement applies to h, but in that case we have uniform C^{m} equivalence for every m.

At any point p ∈ M_{R}, the value of the harmonic map Laplacian Δ_{gh}H at p is an element of T_{H(p)}M, so H_{∗}^{−1} (Δ_{gh}H|_{p}) ∈ T_{p}M. Thus the first component of Θ(v, w) does in fact define a section of T M as claimed.

For this proof, we will denote Möbius coordinates generically by x or (x^{j}) = (x^{1}, …, x^{n+1}), and the associated standard fiber coordinates on T M by v or (v^{j}) = (v^{1}, …, v^{n+1}). Letting E^{j}(x, v) denote the (smooth) component functions of the h-exponential map in Möbius coordinates, we see that H has component functions given by

H^{j}(x) = E^{j}(x, v(x)).

Because E^{j}(x, 0) = x^{j}, it follows from Lemma 2.1(e) with f = E^{j} and u_{0}(x) = (x, 0) that the functions A^{j}(x) = H^{j}(x) − x^{j} satisfy the following uniform bound for sufficiently small v ∈ C̊_{1+λ}^{2,λ}(M_{R}; T M):

∥A^{j}∥_{2,λ;\overline{B}_{1}} ≤ C∥Φ_{i}^{∗}v∥_{2,λ;\overline{B}_{1}} ≤ Cρ(p_{i})^{1+λ}∥v∥_{2,λ,1+λ}.

Calculating in Möbius coordinates, we have

(4.2) (Δ_{gh}H)^{j} = g^{kl} (−∂_{k}∂_{l}H^{j} + Γ_{kl}^{m}∂_{m}H^{j} − (Π_{mq}^{j} ∘ H)∂_{k}H^{m}∂_{l}H^{q}) ,

where Γ_{kl}^{m} are the Christoffel symbols of g and Π_{mq}^{j} are those of h. Note that we can write the difference Γ_{kl}^{m} − Π_{kl}^{m} as follows:

Γ_{kl}^{m} − Π_{kl}^{m} = \frac{1}{2}g^{mj} (∂_{k}g_{lj} + ∂_{l}g_{kj} − ∂_{j}g_{kl})
− \frac{1}{2}h^{mj} (∂_{k}h_{lj} + ∂_{l}h_{kj} − ∂_{j}h_{kl})
= \frac{1}{2}g^{mj} (∂_{k}w_{lj} + ∂_{l}w_{kj} − ∂_{j}w_{kl})
+ \frac{1}{2}(g^{mj} − h^{mj}) (∂_{k}h_{lj} + ∂_{l}h_{kj} − ∂_{j}h_{kl}) .

By virtue of Lemma 2.1 again, this time with f equal to the mjcomponent of the map taking an (n + 1) × (n + 1) matrix to its inverse, this last expression is in C^{0,λ}(\overline{B}_{1}), with C^{1} dependence on w, and satisfies an estimate of the form

∥Γ_{kl}^{m} − Π_{kl}^{m}∥_{0,λ;\overline{B}_{1}} ≤ Cρ(p_{i})^{1+λ}∥w∥_{1,λ,1+λ}.

Now rewrite (4.2) as follows:

(Δ_{gh}H)^{j} = g^{kl}(−∂_{k}∂_{l}A^{j} + (Γ_{kl}^{m} − Π_{kl}^{m})∂_{m}H^{j} + Π_{kl}^{m}∂_{m}A^{j} − Π_{ml}^{j}∂_{k}A^{m}
− Π_{mq}^{j}∂_{k}H^{m}∂_{l}A^{q} + (Π_{mq}^{j} ∘ Id −Π_{mq}^{j} ∘ H)∂_{k}H^{m}∂_{l}H^{q}).

Another application of Lemma 2.1 (with f = Π_{mq}^{j} for the last term) shows that this expression is in C^{0,λ}(\overline{B}_{1}), with C^{1} dependence on v and w, and with C^{0,λ} norm bounded by a multiple of ρ(p_{i})^{1+λ} (∥v∥_{2,λ,1+λ} + ∥w∥_{1,λ,1+λ}). Finally, since the pushforward map H_{∗}^{−1} : T_{H(p)}M → T_{p}M is represented by the inverse of the matrix ∂H^{j}/∂x^{k}(p) = δ_{k}^{j} + ∂A^{j}/∂x^{k}(p), one last application of Lemma 2.1 shows that Θ is a C^{1} map as claimed. q.e.d.

Lemma 4.3. If R is sufficiently small, the differential DΘ_{(0,0)} is a Banach space isomorphism from C̊_{1+λ}^{2,λ}(M_{R}; T M) × C_{1+λ}^{1,λ}(M_{R}; Σ^{2}) to C_{1+λ}^{0,λ}(M_{R}; T M) × C_{1+λ}^{1,λ}(M_{R}; Σ^{2}).

Proof. At (v, w) = (0, 0), the differential of Θ can be computed as follows:

DΘ_{(0,0)}(v, w) = ( \frac{∂}{∂t}|_{t=0} ((H_{tv})_{∗}^{−1}(Δ_{hh}H_{tv})) + \frac{∂}{∂t}|_{t=0} (Δ_{g_{tw}h} Id), w)
= (Lv + Aw, w),

where L is the linearisation of the harmonic map Laplacian Δ_{hh} about the identity map, and A is some first-order linear differential operator that is bounded from C_{1+λ}^{1,λ}(M_{R}; Σ^{2}) to C_{1+λ}^{0,λ}(M_{R}; T M). Clearly this is invertible if and only if L: C̊_{1+λ}^{2,λ}(M_{R}; T M) → C_{1+λ}^{0,λ}(M_{R}; T M) is invertible.

A computation shows that L = ∇_{h}^{∗}∇_{h} − Ric(h). Because Ric(h) approaches −nh at ∂M, it is straightforward to check that (in the terminology of [13]) the characteristic exponents of L are

s = 0, n + 2, \frac{n + 2 ± \sqrt{n^{2} + 8n}}{2}.

It follows that L has indicial radius R = (n + 2)/2, and therefore by [13, Theorem C and Section 7], it is Fredholm as an operator from C_{δ}^{k+2,λ}(M; T M) to C_{δ}^{k,λ}(M; T M) for all k ≥ 0, 0 < λ < 1, and −1 < δ < n + 1. Moreover, [13, Lemma 7.12] shows that for a 1-form u supported in M_{R},

(u, ∇_{h}^{∗}∇_{h}u) ≥ (\frac{n^{2}}{4} + 1 − ε) ∥u∥_{0,2}^{2},

where ε can be made as small as desired by taking R small. Since the operator ∇_{h}^{∗}∇_{h} commutes with the index-raising isomorphism between 1-forms and vector fields, the same result holds for vector fields. It follows that L ∼ ∇_{h}^{∗}∇_{h} + n satisfies an a priori L^{2} estimate of the form

∥v∥_{L^{2}} ≤ C∥Lv∥_{L^{2}} for all v ∈ C_{c}^{∞}(M_{R}; T M)

when R is sufficiently small. Then the same argument as in the proof of Theorem C of [13] implies that L: C̊_{δ}^{k+2,λ}(M_{R}; T M) → C_{δ}^{k,λ}(M_{R}; T M) is an isomorphism for −1 < δ < n + 1; the only modification that needs to be made is to handle the Dirichlet boundary condition on the inner boundary ∂_{R}M_{R}, but as L is uniformly elliptic there, the required estimates follow easily from the standard theory of elliptic boundary value problems. q.e.d.

Now suppose g satisfies the hypotheses of Theorem A, and let w = G^{∗}g −h, where G is given by Lemma 3.2. Let ψ: ℝ → [0, 1] be a smooth cutoff function such that ψ(t) ≡ 1 for t ≤ \frac{1}{2} and ψ(t) ≡ 0 for t ≥ 1. For any small s > 0, define ψ_{s} ∈ C^{∞}(\overline{M}_{R}) by

ψ_{s}(p) = ψ (\frac{ρ(p)}{s}) ,

Then we define w_{s} = ψ_{s}w. Observe that g_{w_{s}} = G^{∗}g on the subset M_{s/2} where ψ_{s} ≡ 1.

Lemma 4.4. For any fixed small R > 0 and any 0 < λ < 1, w_{s} → 0 in C_{1+λ}^{1,λ}(M_{R}; Σ^{2}) as s → 0.

Proof. It is shown in [13, Lemma 3.8] that ψ_{s} is uniformly bounded in C^{1,λ}(M_{R}), with norm independent of s. Because ψ_{s} is supported in M_{s}, it follows from Lemma 2.1(b) that

(4.3)

∥w_{s}∥_{1,λ,1+λ} ≤ C sup_{i} ρ(p_{i})^{−1−λ}∥Φ_{i}^{∗}w_{s}∥_{1,λ;\overline{B}_{2}}

≤ C sup_{\{i:ρ(p_{i})≤s\}} ρ(p_{i})^{−1−λ}∥Φ_{i}^{∗}w∥_{1,λ;\overline{B}_{2}}.

Writing w in background coordinates, we have

w = ρ^{−2}\overline{w}_{jk}(θ, ρ)dθ^{j} dθ^{k},

with \overline{w}_{jk}(θ, ρ) = O(ρ^{2}) and ∂_{l}\overline{w}_{jk}(θ, ρ) = O(ρ). Pulling back in Möbius coordinates, we obtain

Φ_{i}^{∗}w = y^{−2}\overline{w}_{jk}(ρ(p_{i})x + θ(p_{i}), ρ(p_{i})y)dx^{j} dx^{k}.

It follows easily that the component functions of Φ_{i}^{∗}w together with their first and second derivatives are bounded by a constant multiple of ρ(p_{i})^{2}, and so ∥Φ_{i}^{∗}w∥_{1,λ;\overline{B}_{2}} ≤ Cρ(p_{i})^{2}. Inserting this into (4.3) completes the proof. q.e.d.

Theorem 4.5. With g and h as above, for any 0 < λ < 1, there exists a C^{2,λ} collar diffeomorphism Ψ: \overline{M}_{R} → \overline{M} such that Ψ^{∗}g − h ∈ C_{1+λ}^{1,0}(M_{R}; Σ^{2}) and g̃ = Ψ^{∗}g satisfies (1.2) on M_{R_{0}} for some 0 < R_{0} < R.

Proof. Let Θ and w_{s} be defined as above. It follows from Lemma 4.3 and the inverse function theorem that Θ is a bijection from a neighborhood of (0, 0) in C̊_{1+λ}^{2,λ}(M_{R}; T M) × C_{1+λ}^{1,λ}(M_{R}; Σ^{2}) to a neighborhood of (0, 0) in C_{1+λ}^{0,λ}(M_{R}; T M) × C_{1+λ}^{1,λ}(M_{R}; Σ^{2}). By Lemma 4.4, therefore, we can choose s small enough that (0, w_{s}) = Θ(v, w) for some (v, w) ∈ C̊_{1+λ}^{2,λ}(M_{R}; T M) × C_{1+λ}^{1,λ}(M_{R}; Σ^{2}). This is equivalent to the assertion that w = w_{s} and H_{v} is harmonic from (M_{R}, g_{w_{s}}) to (M_{R}, h). Because g_{w_{s}} = G^{∗}g on M_{s/2}, H_{v} is also harmonic from (M_{s/2}, G^{∗}g) to (M_{s/2}, h).

Because locally C^{1} harmonic maps are smooth by elliptic regularity, H_{v}: M_{R} → M_{R} is smooth. Because its component functions H^{j}(x) in Möbius coordinates differ from x^{j} by functions A^{j}(x) that can be made as small as desired in C^{2,λ}(\overline{B}_{2}) (by taking s sufficiently small), it follows that H_{v} is a diffeomorphism from M_{R} to itself, and by Lemma 4.1 it is an admissible map and therefore extends to a homeomorphism of \overline{M}_{R} fixing ∂M pointwise, i.e., a collar diffeomorphism.

Define Ψ = G ∘ H_{v}^{−1} : M_{R} → M, and let g̃ = Ψ^{∗}g. By the diffeomorphism invariance of the Einstein equation, g̃ is Einstein; and by the diffeomorphism invariance of the harmonic map equation, the identity map is harmonic on M_{s/2} from g̃ = (H_{v}^{−1})^{∗} (G^{∗}g) to h. Thus g̃ satisfies (1.2) on M_{s/2}.

Next we will show that Ψ^{∗}g − h ∈ C_{1+λ}^{1,0}(M_{R}; Σ^{2}). Since G^{∗}g − h = w ∈ C_{1+λ}^{1,λ}(M_{R}; Σ^{2}) by the proof of Lemma 4.4, it suffices to show that (H_{v}^{−1})^{∗}G^{∗}g−G^{∗}g ∈ C_{1+λ}^{1,λ}(M_{R}; Σ^{2}). Let us abbreviate G^{∗}g by ĝ and H_{v}^{−1} by K: M_{R} → M_{R}. Again, by taking s (and thus also A^{j}) small enough, we can ensure that H_{v}(V_{2}(p_{i})) contains \overline{V}_{1}(p_{i}) for each Möbius chart, and therefore K is an admissible map. We will write the component functions of H_{v} as H^{j}(x) = x^{j} + A^{j}(x) as above, with

∥A^{j}∥_{2,λ;\overline{B}_{2}} ≤ Cρ(p_{i})^{1+λ}.

If s is chosen small enough, then the functions A^{j} will be uniformly small in C^{2,λ}(\overline{B}_{2}), so the Jacobian matrix ∂_{k}H^{j}(x) will be uniformly invertible on \overline{B}_{2}, independent of the choice of Möbius chart. Let us write (I_{k}^{j}(x)) for the components of the inverse matrix of (∂_{k}H^{j}(x)). Because matrix inversion is continuous in the C^{1,λ} norm by Lemma 2.1(d), the functions I_{k}^{j} are uniformly bounded in C^{1,λ}(\overline{B}_{2}).

The chain rule shows that

(4.4)  ∂_{k}K^{j}(x) = I_{k}^{j}(K(x)),

which is continuous and uniformly bounded on \overline{B}_{1}. But this implies that K^{j} is uniformly C^{1}, and using (4.4) we conclude successively that ∂_{k}K^{j}(x) is uniformly C^{1}, K^{j} is uniformly C^{2}, ∂_{k}K^{j}(x) is uniformly C^{1,λ} (by Lemma 2.1(c)), and thus K^{j} is uniformly C^{2,λ} on \overline{B}_{1}.

Let us write B^{j}(x) = K^{j}(x)−x^{j}. The fact that H_{v}∘K = Id translates in Möbius coordinates to

K^{j}(x) + A^{j}(K(x)) = x^{j},

so B^{j}(x) = −A^{j}(K(x)), and applying Lemma 2.1(c) one more time, we conclude that

(4.5) ∥B^{j}∥_{2,λ;\overline{B}_{1}} = ∥A^{j} ∘ K∥_{2,λ;\overline{B}_{1}} ≤ C∥A^{j}∥_{2,λ;\overline{B}_{2}} ≤ C′ρ(p_{i})^{1+λ}.

Now to show that K^{∗}ĝ − ĝ ∈ C_{1+λ}^{1,0}(M_{R}; Σ^{2}), we just compute in Möbius coordinates:

K^{∗}ĝ − ĝ = ĝ_{jk}(K(x))(dx^{j} + dB^{j})(dx^{k} + dB^{k}) − ĝ_{jk}(x)dx^{j} dx^{k}
= (ĝ_{jk}(K(x)) − ĝ_{jk}(x))dx^{j} dx^{k}

+ 2ĝ_{jk}(K(x))\frac{∂B^{k}}{∂x^{q}} dx^{j} dx^{q} + ĝ_{jk}(K(x))\frac{∂B^{j}}{∂x^{m}} \frac{∂B^{k}}{∂x^{q}} dx^{m} dx^{q}.

Because ρ^{2}ĝ ∈ C^{2}(\overline{M}), the component functions ĝ_{jk} are uniformly bounded in C^{2}(\overline{B}_{2}) by the properties of Möbius coordinates. The same is true of ĝ_{jk} ∘ K by composition. The last two terms above thus have C^{1} norms uniformly bounded by a constant multiple of ρ(p_{i})^{1+λ}. Differentiating the first term, we obtain

∂_{l} (ĝ_{jk} ∘ K − ĝ_{jk}) = (∂_{m}ĝ_{jk} ∘ K) \frac{∂K^{m}}{∂x^{l}} − ∂_{l}ĝ_{jk}

= (∂_{m}ĝ_{jk} ∘ K) \frac{∂B^{m}}{∂x^{l}} + (∂_{l}ĝ_{jk} ∘ K − ∂_{l}ĝ_{jk}) .

The first term is uniformly bounded by a multiple of ρ(p_{i})^{1+λ} thanks to (4.5). The same is true of the second term by a simple application of the mean value theorem and the fact that K^{j}(x) − x^{j} = B^{j}(x). This completes the proof that Ψ^{∗}g − h ∈ C_{1+λ}^{1,0}(M_{R}; Σ^{2}).

It remains only to show that Ψ has a C^{2,λ} extension to \overline{M}_{R}. Because G is C^{3} on \overline{M}_{R} by construction, it suffices to consider H_{v}^{−1} = K. Choose a fixed p_{i} ∈ M_{R} and corresponding Möbius chart Φ_{i}. We can write the map K either in Möbius coordinates, with coordinate functions denoted by (K^{j}) as above:

K(x^{j}) = (K^{1}(x^{j}), …, K^{n+1}(x^{j})) ,

or in background coordinates (θ^{j}) = (θ^{1}, …, θ^{n}, ρ), with coordinate functions that we will denote by (\overline{K}^{j}):

K(θ^{j}) = (\overline{K}^{1}(θ^{j}), …, \overline{K}^{n+1}(θ^{j})) .

The two coordinate representations are related by

\overline{K}^{m}(θ^{j}) = c^{m} + ρ(p_{i})K^{m} (\frac{θ^{j} − c^{j}}{ρ(p_{i})} ) ,

where c^{j} are constants defined by (c^{1}, …, c^{n+1}) = (θ^{1}(p_{i}), …, θ^{n}(p_{i}), 0). Using once again the fact that K^{j}(x) = x^{j} + B^{j}(x) with B^{j} satisfying (4.5), we compute

\frac{∂\overline{K}^{m}}{∂θ^{k}} = δ_{k}^{m} + \frac{∂B^{m}}{∂x^{k}} (\frac{θ^{j} − c^{j}}{ρ(p_{i})} ) ,

so both \overline{K}^{m} and ∂\overline{K}^{m}/∂x^{k} are uniformly bounded. Differentiating once more, we find

\frac{∂^{2}\overline{K}^{m}}{∂θ^{k}∂θ^{l}} = ρ(p_{i})^{−1} \frac{∂^{2}B^{m}}{∂x^{k}∂x^{l}} (\frac{θ^{j} − c^{j}}{ρ(p_{i})} ) ,

and therefore,

|\frac{∂^{2}\overline{K}^{m}}{∂θ^{k}∂θ^{l}} (θ^{j}) − \frac{∂^{2}\overline{K}^{m}}{∂θ^{k}∂θ^{l}} (θ̃^{j})|

= ρ(p_{i})^{−1} |\frac{∂^{2}B^{m}}{∂x^{k}∂x^{l}} (\frac{θ^{j} − c^{j}}{ρ(p_{i})} ) − \frac{∂^{2}B^{m}}{∂x^{k}∂x^{l}} (\frac{θ̃^{j} − c^{j}}{ρ(p_{i})} )|

≤ ρ(p_{i})^{−1}∥B^{m}∥_{2,λ;\overline{B}_{2}} |\frac{θ^{j} − θ̃^{j}}{ρ(p_{i})} |^{λ}

≤ C |θ^{j} − θ̃^{j}|^{λ} ,

which shows that \overline{K}^{m} is uniformly C^{2,λ} up to the boundary as claimed. q.e.d.

5. Polyhomogeneity

Let U_{0} ⊂ ℝ^{n} be an open set, and let U = U_{0} × (0, ε) ⊂ ℍ^{n+1}. For any δ ∈ ℝ, we denote by 𝒞^{δ} the space of functions f ∈ C^{∞}(U) that satisfy, on any subset K × (0, ε_{0}) with K ⊂ U_{0} compact and 0 < ε_{0} < ε, estimates of the following form for all integers r ≥ 0 and all multi-indices α:

|(y∂_{y})^{r}∂_{x}^{α}f(x, y)| ≤ C_{r,α}y^{δ}.

(We use the multi-index notations α = (α_{1}, …, α_{n}) and ∂_{x}^{α} = (∂_{x^{1}})^{α_{1}} … (∂_{x^{n}})^{α_{n}}.)

A smooth function f: U → ℝ is said to be polyhomogeneous (cf. [3, 16]) if there exists a sequence of real numbers s_{i} → +∞, a sequence of nonnegative integers {q_{i}}, and functions f_{ij} ∈ C^{∞}(U_{0}) such that

(5.1)  f(x, y) ∼ ∑_{i=1}^{∞} ∑_{j=0}^{q_{i}} y^{s_{i}}(log y)^{j}f_{ij}(x)

in the sense that for any δ > 0, there exists a positive integer N such that

f(x, y) − ∑_{i=1}^{N} ∑_{j=0}^{q_{i}} y^{s_{i}}(log y)^{j}f_{ij}(x) ∈ 𝒞^{δ}.

A function or tensor field on M_{R} is said to be polyhomogeneous if its coordinate representation in every background chart is polyhomogeneous. (Note that the definition of polyhomogeneity is phrased somewhat differently in [3], but it is easy to verify that the two definitions are equivalent. Note also that there is a misprint in the first displayed inequality of [3, Section 3.1.5]: ∂_{y}^{α} in that inequality should be replaced by ∂_{v}^{α}.)

In this section, we will apply the theory of [3] to conclude that solutions to (1.2) are polyhomogeneous. A key step in the proof will be a regularity result for the linearised operator D_{1}Q_{(h,h)} = \frac{1}{2}(Δ_{L} + 2n) from [13]. Following [3], we say that an interval (δ_{−}, δ_{+}) ⊂ ℝ is a (weak) regularity interval for a second-order linear operator P on the spaces C_{δ}^{k,λ}(M_{R}; Σ^{2}) if whenever u is a locally C^{2} section of Σ^{2} such that u ∈ C_{δ_{0}}^{0,0}(M_{R}; Σ^{2}) and P u ∈ C_{δ}^{0,λ}(M_{R}; Σ^{2}) with λ ∈ (0, 1) and δ_{−} < δ_{0} < δ < δ_{+}, it follows that u ∈ C_{δ}^{2,λ}(M_{R}; Σ^{2}). (We caution the reader that the notations for weighted Hölder spaces used in [3] are different from those of [13] that we are using here, in terms of index positions and normalization of weights. The space of sections of Σ^{2} that is denoted by C_{k+λ}^{δ}(M) in [3] is equal to the space that we would call C_{δ+2}^{k,λ}(M_{R}; Σ^{2}). The difference between the weight factors δ and δ + 2 arises because we measure the size of the component functions in Möbius coordinates, while [3] measures them in background coordinates. To avoid confusion, we will use the notation AC_{k+λ}^{δ}(M) for the space denoted by C_{k+λ}^{δ}(M) in [3], so that AC_{k+λ}^{δ}(M) = C_{δ+2}^{k,λ}(M_{R}; Σ^{2}). The condition that we have defined here would be expressed in [3] by saying that (δ_{−} − 2, δ_{+} − 2) is a regularity interval for P on the spaces AC_{k+λ}^{δ}(M).)

Theorem 5.1. With g̃ = Ψ^{∗}g as in the preceding section, g̃ is polyhomogeneous.

Proof. For any small symmetric 2-tensor φ on M_{R}, define

F [φ] := ρ^{2}Q(h + ρ^{−2}φ, h),

with Q as in (1.2). Then φ = ρ^{2}(g̃ − h) satisfies F [φ] = 0. We wish to apply [3, Theorem 5.1.1] to F, and thereby conclude that φ is polyhomogeneous. To do so, we will consider φ_{0} ≡ 0 as an approximate solution to F [φ] = 0, and check that F, φ, and φ_{0} satisfy each of the hypotheses of Theorem 5.1.1.

(i) F is a geometric operator in the sense of [3]: This follows easily from the fact that F is an invariant operator on tensor fields. (Note that the notion of geometricity defined in [3] is considerably weaker than that of [13]; in particular, the definition in [3] does not require the coordinate expression of F [φ] to depend only on the coefficients of a single metric and the covariant derivatives of its curvature.)

(ii) F is quasilinear, and F [φ] can be written in background coordinates as a smooth function of (ρ, θ^{α}, φ_{ij}, ρ∂_{k}φ_{ij}, ρ^{2}∂_{k}∂_{l}φ_{ij}): This is easily seen by expanding Q(g̃, h) in background coordinates.

(iii) F [φ_{0}] is a smooth tensor field on \overline{M} and an element of AC_{0}^{δ_{0}}(M) = C_{δ_{0}+2}^{0,0}(M; Σ^{2}) for some δ_{0} > 1: Observe that F [φ_{0}] = F [0] = ρ^{2}Q(h, h) = ρ^{2}(Ric(h) + nh). Using the formula for the transformation of the Ricci tensor under a conformal change of metric (cf. [4, p. 59]) this can be written in background coordinates as

F [φ_{0}] = ρ^{2} (R_{ij} + nh_{ij})

= ρ^{2}\overline{R}_{ij} + (n − 1)ρρ_{;ij} + ρ\overline{h}^{kl}ρ_{;kl}\overline{h}_{ij} − n\overline{h}^{kl}ρ_{;k}ρ_{;l}\overline{h}_{ij} + n\overline{h}_{ij},

where the curvature and covariant derivatives are computed with respect to the smooth metric \overline{h} = ρ^{2}h. Because of the way we constructed \overline{h}, |dρ|2 ; \overline{h} is identically equal to 1 and ρ_{;ij} is identically zero, so all terms after the first one cancel, showing that F [φ_{0}] = ρ^{2}\overline{R}_{ij}, which is smooth on \overline{M} and O(ρ^{2}) in background coordinates, and thus an element of AC_{0}^{2}(M).

(iv) φ − φ_{0} ∈ AC_{1}^{δ}(M_{R}) = C_{δ+2}^{1,0}(M_{R}; Σ^{2}) for some δ > 1 and R > 0: Since φ − φ_{0} = φ = ρ^{2}(g̃− h), this is equivalent to the assertion that g̃ − h ∈ C_{δ}^{1,0}(M_{R}; Σ^{2}) for some δ > 1, which is guaranteed by Theorem 4.5.

(v) The linearised operator F′[φ_{0}] is a geometric elliptic operator satisfying conditions (4.2.1)–(4.2.4) of [3]: Actually, this is not true as stated, but something just as good is true. Note that F′[φ_{0}] = ρ^{2} ∘ D_{1}Q_{(h,h)} ∘ ρ^{−2} = ρ^{2} ∘ (Δ_{L} + 2n) ∘ ρ^{−2}, which is certainly a geometric elliptic operator.

Define subbundles of Σ^{2} as follows:

V_{0} = span(g);
V_{1} = {q ∈ Σ^{2} : Tr_{g} q = 0, q(grad ρ,·) = 0};
V_{2} = span (g − \frac{n + 1}{|dρ|_{g}^{2}}dρ ⊗ dρ);
V_{3} = {dρ ⊗ ω + ω ⊗ dρ : ω ∈ T^{∗}M, ⟨ω, dρ⟩ = 0} .

It is easy to check that Σ^{2} admits an orthogonal decomposition Σ^{2} = V_{0} ⊕ V_{1} ⊕ V_{2} ⊕ V_{3}. For i = 0, …, 3, let π_{i}: Σ^{2} → V_{i} denote the orthogonal projection. The arguments of [9, pp. 199–202] show that Δ_{L} + 2n can be written in the form

Δ_{L} + 2n = ∑_{i=0}^{3} P_{i} + P̃,

where in background coordinates P̃ is of the form [3, (4.2.3)] and each P_{i} is an operator on sections of V_{i} that can be written

P_{i} = − (ρ^{2}∂_{ρ}^{2} + (5 − n)ρ∂_{ρ} + B_{i}) ⊗ π_{i},

with B_{0} = B_{2} = 4 − 2n, B_{1} = 4 − 4n, and B_{3} = 3 − 3n. Thus

F′[φ_{0}] = ∑_{i=0}^{3} L_{i} + L̃,

where L̃ = ρ^{2} ∘ P̃ ∘ ρ^{−2} is again of the form [3, (4.2.3)], and

L_{i} = ρ^{2} ∘ P_{i} ∘ ρ^{−2} = − (ρ^{2}∂_{ρ}^{2} + (1 − n)ρ∂_{ρ} + b_{i}) ⊗ π_{i},

with b_{i} = B_{i}−4 < 0. All of the arguments in [3, Sections 4 and 5] go through with only trivial changes if the ordinary differential operator denoted there by L_{ab} is replaced by the block-diagonal operator L_{0} ⊕ L_{1} ⊕ L_{2} ⊕ L_{3}.

(vi) The interval (0, n) is a regularity interval for the operator F′[φ_{0}] on the spaces AC_{k+λ}^{δ}(M_{R}): It is an immediate consequence of Lemma 6.4(b) of [13] that (0, n) is a regularity interval for Δ_{L} + 2n on the spaces C_{δ}^{k,λ}(M; Σ_{R}), and it follows immediately from this that (−2, n − 2) is a regularity interval for F′[φ_{0}] = ρ^{2} ∘ (Δ_{L} + 2n) ∘ ρ^{−2} on the same spaces. In the terminology of [3], this means that (0, n) is a regularity interval on AC_{k+λ}^{δ}(M_{R}).

¿From [3, Theorem 5.1.1], therefore, we conclude that φ (and hence also g̃) is polyhomogeneous. q.e.d.

6. The asymptotic expansion

To obtain the asymptotic expansion announced in Theorem A, we need to subject g to one more collar diffeomorphism. First, a preliminary lemma.

Lemma 6.1. Suppose g̃ is a Riemannian metric on M_{R} that is polyhomogeneous and conformally compact of class C^{1,λ} for some 0 < λ < 1, and satisfies |dρ|_{ρ^{2}g̃} → 1 at ∂M. Then there exists a polyhomogeneous C^{1,λ} defining function r such that |dr|_{r^{2}g̃} ≡ 1 in a neighborhood of ∂M and r/ρ → 1 at ∂M.

Proof. Writing \overline{g} = ρ^{2}g̃ and r = ρe^{u}, we see that the conclusion is equivalent to

|dρ|2 ; \overline{g} + 2ρ⟨dρ, du⟩\overline{g} + ρ^{2}|du|2 ; \overline{g} = 1, u|_{∂M} = 0.

It is shown in [12, Lemma 5.1] that this has a solution u ∈ C^{2,λ}(\overline{M}_{R}; ℝ) if \overline{g} is C^{3,λ} up to the boundary, by reducing it to finding the flow of the Hamiltonian vector field X_{F}, where F: T^{∗}\overline{M}_{R} → ℝ is the function defined by

F (θ, ξ) = 2⟨dρ, ξ⟩\overline{g} + ρ|ξ|2 ; \overline{g} − \frac{1 − |dρ|2 ; \overline{g}}{ρ} .

(Here (θ, ξ) = (θ^{1}, …, θ^{n+1}, ξ_{1}, …, ξ_{n+1}) are standard coordinates on T^{∗}\overline{M}_{R} associated with background coordinates.) In the present situation, F is only of class C^{0,λ} on T^{∗}\overline{M}_{R}; but because it is polyhomogeneous, each of the following quantities is also C^{0,λ} in each background chart:

\frac{∂F}{∂ξ_{j}} , \frac{∂F}{∂θ^{α}} , ρ\frac{∂F}{∂ρ} ,

\frac{∂^{2}F}{∂ξ_{j}∂ξ_{k}} , \frac{∂^{2}F}{∂θ^{α}∂θ^{β}} , \frac{∂^{2}F}{∂ξ_{j}∂θ^{α}} , ρ\frac{∂^{2}F}{∂θ^{α}∂ρ}, ρ\frac{∂^{2}F}{∂ξ_{j}∂ρ}, ρ^{2} \frac{∂^{2}F}{∂ρ^{2}} .

Since the normal component of X_{F} satisfies

dρ(X_{F})|_{∂T^{∗}\overline{M}_{R}} = \frac{∂F}{∂ξ_{n+1}} |_{∂T^{∗}\overline{M}_{R}} = 2\overline{g}^{n+1,n+1}|_{∂T^{∗}\overline{M}_{R}} = 2,

it follows that V = \frac{1}{2}X_{F} satisfies the hypotheses of Lemma 6.3 below, so the flow-out by X_{F} from the boundary of T^{∗}\overline{M}_{R} exists and is polyhomogeneous and of class C^{0,λ}. The rest of the argument in [12, Lemma 5.1] then goes through to prove the existence of r as claimed. Moreover, the solution r so obtained is itself polyhomogeneous; matching lowestorder terms in the expansion of r with those in the expansion of \overline{g}, we find that r ∈ C^{1,λ}(\overline{M}_{R}) as claimed. q.e.d.

It is clear from the proof of the preceding lemma that near any boundary point, one can choose new coordinates in which r is one of the coordinate functions and the metric is polyhomogeneous. In fact, we can do much better, as the next lemma shows.

Lemma 6.2. If g̃ satisfies the hypotheses of the preceding lemma, then for R sufficiently small, there exists a polyhomogeneous C^{1,λ} collar diffeomorphism Γ: \overline{M}_{R} → \overline{M} such that Γ^{∗}g̃ has the form (1.1).

Proof. Let r be the polyhomogeneous C^{1,λ} defining function given by Lemma 6.1. As in the proof of Lemma 3.2, let P be the r^{2}g̃-gradient of r, which is a polyhomogeneous C^{0,λ} vector field on \overline{M}_{R} whose normal component satisfies dρ(P) = 1 along ∂M. Using Lemma 6.3 again, in each background coordinate chart Ω we obtain a uniquely determined C^{0,λ} polyhomogeneous flow (x, t) ↦ γ_{x}(t), where for each x ∈ Ω ∩ ∂M, γ_{x}: [0, ε] → \overline{M}_{R} is the integral curve of P starting at x. Comparing lowest-order terms in the expansions of γ_{x}(t) and P, we see that the flow is in fact C^{1,λ} up to the boundary. The various maps thus obtained in different coordinate charts all agree where they overlap, so they patch together to define a global map Γ: ∂M × [0, ε] → \overline{M}_{R}. The inverse function theorem shows that Γ is a diffeomorphism in a neighborhood of ∂M × \{0\}. Identifying ∂M × [0, ε] with \overline{M}_{ε}, we can view Γ as a collar diffeomorphism. It is easy to check that Γ^{∗}g̃ has the form (1.1) with G(ρ) polyhomogeneous. q.e.d.

Here is the ODE lemma used in the proofs of Lemmas 6.1 and 6.2. It is adapted from Proposition B.1 in [6]. In this lemma, ℍ^{m+1} denotes the upper half-space in ℝ^{m+1}, with coordinates (x^{1}, …, x^{m}, y). In our application of this lemma in the proof of Lemma 6.1, the x^{i}-coordinates correspond to (θ^{1}, …, θ^{n}, ξ_{1}, …, ξ_{n+1}), and y corresponds to ρ = θ^{n+1}.

Lemma 6.3. Let U_{0} be an open subset of ℝ^{m}, and let V be a C^{1} vector field on U = U_{0} × (0, ε) ⊂ ℍ^{m+1} of the form

V = A^{i}(x, y)\frac{∂}{∂x^{i}} + (1 + B(x, y))\frac{∂}{∂y} .

Suppose that A^{i} and B satisfy the following estimates for some constants C_{0} > 0 and 0 < λ < 1:

(6.1)  \begin{aligned}|B|, |∂_{x^{j}}B|, |y∂_{y}B| & ≤ C_{0}y^{λ}, \\ |A^{i}|, |∂_{x^{j}}A^{i}|, |y∂_{y}A^{i}| & ≤ C_{0}y^{λ−1}. \end{aligned}

(a) If K is any compact subset of U_{0}, there exists ε > 0 such that, for each x_{0} ∈ K, there is a unique continuous solution γ = γ_{x_{0}} on [0, ε] to the initial value problem

(6.2)  \begin{aligned}γ′(t) & = V (γ(t)), \\ γ(0) & = (x_{0}, 0). \end{aligned}

(b) If the coefficient functions A^{i} and B are polyhomogeneous, then the map (x, t) ↦ γ_{x}(t) is polyhomogeneous and C^{0,λ} up to the boundary.

Proof. Let x_{0} ∈ K be arbitrary. Choose constants C_{1} > 0 and 0 < α < λ, and let ε be a positive constant to be determined later. Let 𝒳 denote the set of continuous maps γ: [0, ε] → ℝ^{m+1} of the form

(6.3)  γ(t) = (x_{0} + a(t), t + b(t))

satisfying

(6.4)  |a^{i}(t)| ≤ C_{1}t^{α},
(6.5)  |b(t)| ≤ C_{1}t^{α+1}.

If ε is sufficiently small, all such maps take their values in U for t ∈ (0, ε], and 𝒳 is a complete metric space when endowed with the metric

d(γ, γ̃) = sup_{t∈[0,ε]} (t^{−α}|a(t) − ã(t)| + t^{−α−1}|b(t) − b̃(t)|) .

Note that for any γ ∈ 𝒳 , we have |y(γ(t)) − t| = |b(t)| ≤ C_{1}t^{1+α} ≤ C_{1}ε^{α}t, so choosing ε small enough that C_{1}ε^{α} < \frac{1}{2} implies that

(6.6)  \frac{1}{2}t ≤ y(γ(t)) ≤ \frac{3}{2}t.

Define a map T: 𝒳 → 𝒳 by

T γ(t) = (x_{0}^{i} + ∫_{0}^{t} A^{i}(γ(τ)) dτ, t + ∫_{0}^{t} B(γ(τ)) dτ) .

It is straightforward to check that

∫_{0}^{t} |A^{i}(γ(τ))| dτ ≤ ∫_{0}^{t} C_{0}y(γ(t))^{λ−1} dτ
≤ ∫_{0}^{t} C_{0}Cτ^{λ−1} dτ ≤ C′t^{λ} ≤ C′ε^{λ−α}t^{α};
∫_{0}^{t} |B(γ(τ))| dτ ≤ ∫_{0}^{t} C_{0}y(γ(t))^{λ} dτ
≤ ∫_{0}^{t} C_{0}Cτ^{λ} dτ ≤ C′t^{λ+1} ≤ C′ε^{λ−α}t^{α+1}.

Thus if we choose ε small enough, it follows that T maps 𝒳 into 𝒳 .

We will show that, after choosing ε even smaller if necessary, T is a contraction. Suppose γ, γ̃ ∈ 𝒳 . If (x^{∗}, y^{∗}) is any point along the line segment between γ(t) and γ̃(t), (6.6) implies that \frac{1}{2}t ≤ y^{∗} ≤ \frac{3}{2}t. To estimate d(T γ, T γ̃), we use the mean-value theorem to obtain

t^{−α} ∫_{0}^{t}|A^{i}(γ(τ)) − A^{i}(γ̃(τ))| dτ

≤ t^{−α} ∑_{j} ∫_{0}^{t} |∂_{x^{j}}A^{i}(x^{∗}, y^{∗})| |a^{j}(τ) − a^{j}(τ)| dτ

+ t^{−α} ∫_{0}^{t} |∂_{y}A^{i}(x^{∗}, y^{∗})| |b(τ) − b̃(τ)| dτ

≤ t^{−α} ∫_{0}^{t} mC_{0}(y^{∗})^{λ−1}τ^{α}d(γ, γ̃) dτ

+ t^{−α} ∫_{0}^{t} C_{0}(y^{∗})^{λ−2}τ^{α+1}d(γ, γ̃) dτ

≤ t^{−α} ∫_{0}^{t} C_{0}Cτ^{λ+α−1}d(γ, γ̃) dτ

≤ C′ε^{λ}d(γ, γ̃),

for some (x^{∗}, y^{∗}) on the line segment between γ(τ) and γ̃(τ). The analogous estimate for the y-component is similar. Thus T is a contraction if we choose ε sufficiently small. It therefore has a unique fixed point in 𝒳 , which is a solution to (6.2). By compactness, it is clear that for any given C_{1} we can choose ε uniformly for x_{0} ∈ K.

To see that the solution is unique, suppose γ is any continuous solution to (6.2). If we write γ in the form (6.3), the equation T γ = γ together with (6.1) implies successively that |b(t)| ≤ Ct, then |b(t)| ≤ Ct^{λ+1}, and finally |a^{i}(t)| ≤ Ct^{λ}. It follows that (6.4) and (6.5) hold on [0, ε] if C_{1} and ε are chosen appropriately. Thus the restriction of γ to [0, ε] is in 𝒳 , and so is equal to the unique fixed point of T.

Finally, we will prove the polyhomogeneity of the solution when V is polyhomogeneous. Recall the spaces 𝒞^{δ} defined at the beginning of Section 5. We will need the following fact about these spaces, which is proved by a straightforward analysis of the Taylor expansion of F (u + f, v + g) about (u, v) = (u^{1}, …, u^{m}, v):

(6.7) \begin{matrix}F (u(x, y) + f(x, y), v(x, y) + g(x, y)) − F (u(x, y), v(x, y)) \\ ∈ 𝒞^{δ+γ} \\ when F ∈ 𝒞^{δ}, u^{i} ∈ 𝒞^{0}, f^{i} ∈ 𝒞^{γ}, v ∈ 𝒞^{1}, g ∈ 𝒞^{γ+1}, γ ≥ 0. \end{matrix}

For this proof, let 𝒜 denote the space of polyhomogeneous functions on U, and for any δ ∈ ℝ, define 𝒜^{δ} = 𝒜 ∩𝒞^{δ}. Thus a polyhomogeneous function f is in 𝒜^{δ} if and only if its leading term in the expansion (5.1) satisfies s_{1} ≥ δ and, if s_{1} = δ, q_{1} = 0. Our hypotheses imply that A^{i} ∈ 𝒜^{λ−1} and B ∈ 𝒜^{λ}.

For each x_{0} ∈ K, let a(x_{0}, t) and b(x_{0}, t) denote the functions a(t) and b(t) obtained above with initial condition (x_{0}, 0). The standard argument showing that solutions of ODEs depend smoothly upon initial values can be used to obtain estimates of the following form for all multiindices α:

|∂_{x}^{α}a^{i}(x, t)| ≤ C_{α}t^{λ}, |∂_{x}^{α}b(x, t)| ≤ C_{α}t^{λ+1}.

It is then straightforward to use the differential equation to obtain estimates on (t∂_{t})^{r}∂_{x}^{α}a^{i}(x, t) and (t∂_{t})^{r}∂_{x}^{α}b(x, t), showing that a^{i} ∈ 𝒞^{λ} and b ∈ 𝒞^{λ+1}.

Suppose that for some integer m ≥ 1, we have a “partial polyhomogeneous expansion” of the form

(6.8)  \begin{matrix}a^{i}(x, t) = p^{i}(x, t) + r^{i}(x, t),  b(x, t) = q(x, t) + s(x, t), \\ with p^{i} ∈ 𝒜^{λ}, r^{i} ∈ 𝒞^{mλ}, q ∈ 𝒜^{λ+1}, s ∈ 𝒞^{mλ+1}. \end{matrix}

The discussion in the preceding paragraph shows that (6.8) holds with m = 1 and p^{i} = q = 0. Inserting (6.8) into T γ = γ, we obtain r^{i} = r_{0}^{i} + r_{1}^{i}, where

r_{0}^{i} = ∫_{0}^{t} A^{i}(x + p(x, τ), τ + q(x, τ)) dτ − p^{i}(x, t),
r_{1}^{i} = ∫_{0}^{t}(A^{i}(x + p(x, τ) + r(x, τ), τ + q(x, τ) + s(x, τ))
− A^{i}(x + p(x, τ), τ + q(x, τ))) dτ.

¿From (6.7) we conclude that r_{1}^{i} ∈ 𝒞^{(m+1)λ}. On the other hand, it is easy to check that r_{0}^{i} is polyhomogeneous, and since r_{0}^{i} = r^{i} −r_{1}^{i} ∈ 𝒞^{mλ}, we have r_{0}^{i} ∈ 𝒜^{mλ}. A similar argument shows that s = s_{0} + s_{1}, with s_{0} ∈ 𝒜^{mλ+1} and s_{1} ∈ 𝒞^{(m+1)λ+1}. We let P^{i} denote the sum of the (finitely many) terms in the expansion of r_{0}^{i} that are in 𝒜^{mλ} ∖𝒞^{(m+1)λ}, and Q the sum of the terms in s_{0} that are in 𝒜^{mλ+1} ∖ 𝒞^{(m+1)λ+1}. Replacing p^{i} by p^{i} + P^{i} and q by q + Q, we obtain (6.8) with m + 1 in place of m. Continuing by induction, we conclude that a and b are polyhomogeneous. Since both are in 𝒜^{λ}, it follows that (x, t) ↦ γ_{x}(t) is a C^{0,λ} map. q.e.d.

We are finally ready to prove the main theorem.

Proof of Theorem A. Suppose g satisfies the hypotheses of the theorem. Let Ψ be the collar diffeomorphism given by Theorem 4.5, and let g̃ = Ψ^{∗}g, which is polyhomogeneous and conformally compact of class C^{1,λ}, 0 < λ < 1. Theorem 4.5 shows that ρ^{2}Ψ^{∗}g − ρ^{2}h ∈ C_{3+λ}^{1,0}(M_{R}; Σ^{2}), which implies that ρ^{2}Ψ^{∗}g − ρ^{2}h = O(ρ^{1+λ}) in background coordinates, so Ψ^{∗}g and h have the same conformal infinity γ̃.

Then let Γ be the collar diffeomorphism given by Lemma 6.2, so that Γ^{∗}g̃ has the form (1.1). Because r/ρ → 1 at ∂M, it follows that Γ^{∗}g̃ also has γ̃ as conformal infinity. Because ρ^{2}Γ^{∗}g̃ is continuous up to ∂M and has a smooth restriction to ∂M, it follows that the log terms in the asymptotic expansion for G(ρ) all occur with positive powers of ρ. Once polyhomogeneity is known, the detailed form of the expansion is established by matching powers and coefficients of various terms appearing in the equations. Such a study has been done by Robin Graham and Charles Fefferman [8] (see also [11]), and the results there imply that the expansions are of the form described in Theorem A. In the special case when dim M = 3, it is possible to choose the conformal infinity γ̃ to have constant Gaussian curvature, and an easy computation shows that the first log term vanishes in this case (cf. [7, 8, 11]), so ρ^{2}Γ^{∗}g̃ is always smooth.

The proof is completed by letting Φ be the collar diffeomorphism Ψ ∘ Γ. q.e.d.

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Piotr T. Chruściel, Département de mathématiques, Faculté des Sciences, Parc de Grandmont, F37200 Tours, France

E-mail address: piotr@gargan.math.univ-tours.fr
URL: http://www.phys.univ-tours.fr/~piotr

Erwann Delay, Département de mathématiques, Faculté des Sciences, rue Louis Pasteur, F84000 Avignon, France

E-mail address: Erwan.Delay@univ-avignon.fr
URL: http://www.phys.univ-tours.fr/~delay

John M. Lee, University of Washington, Department of Mathematics, Box 354350, Seattle, WA 98195-4350

E-mail address: lee@math.washington.edu
URL: http://www.math.washington.edu/~lee

Dale N. Skinner, Milliman USA, 1301 Fifth Avenue, Suite 3800, Seattle, WA 98101

E-mail address: dales82@mindspring.com
URL: http://www.daleskinner.com