On the Regularity of Solutions to the Yamabe Equation and the Existence of Smooth Hyperboloidal

Initial Data for Einstein’s Field Equations

Lars Andersson^{∗} Department of Mathematics Royal Institute of Technology S-100 44 Stockholm, Sweden

Piotr T. Chruściel^{†}

Center for Mathematics and its Applications Australian National University Canberra ACT2601, Australia

Helmut Friedrich

Max-Planck Institut für Astrophysik

Karl-Schwarzschild-Str. 1 D-8046 Garching bei München, FRG

September 6, 1999

Abstract

The regularity of the solutions to the Yamabe Problem is considered in the case of conformally compact manifolds and negative scalar curvature.The existence of smooth hyperboloidal initial data for Einstein’s field equations is demonstrated.

1 Introduction

In this paper we shall show the existence of smooth hyperboloidal initial data for Einstein’s field equations. Such data occur under the following circumstances. Suppose one is given a solution to Einstein’s source free field equations (of signature (–,+,+,+)) with cosmological constant Λ ≤ 0 which is ”asymptotically flat” resp.”asymptotically simple” in the sense of Penrose [15]. Let M̃ be a space-like hypersurface in this space-time which extends up to the conformal boundary I. The latter is a null hypersurface if Λ = 0 and time-like if Λ < 0. If one assumes that M̃ can be extended across I as a smooth space-like hypersurface in some smooth conformal extension of the given space-time and denotes by ∂M the intersection of this extension with I, then M = M̃ ∩ ∂M will be a smooth compact manifold with boundary.

∗Supported in part by NFR, the Swedish Academy of Sciences and the Gustavsson foundation.

†On leave from the Institute of Mathematics of the Polish Academy of Sciences, Warsaw.

We call the tripel (M̃, g̃, χ̃), where g̃ is the Riemannian metric and χ̃ the second fundamental form induced on M̃ by the space-time metric, a “hyperboloidal initial data set”. Since this initial data set has been obtained from a solution to the Einstein equations the fields g̃ and χ̃ satisfy the constraint equations. Because of the specific geometric situation which is represented by this data set and because it has been derived from a space-time admitting a smooth conformal boundary I at infinity the fields g̃ and χ̃ satisfy certain special smoothness resp. fall-off conditions at ∂M.

The first and most fundamental of these is an immediate consequence of the definition of the conformal boundary I. There exist on M a smooth Riemannian metric h and a smooth defining function Ω of ∂M such that

g̃ = Ω^{−2}h on M̃  (1.1)

The second condition is more complicated to analyse and to explain. For this reason we study in this paper the class of data satisfying the condition

χ̃ = \frac{1}{3}trg̃(χ̃)g̃.  (1.2)

This is similar to the condition of time-symmetry for standard Cauchy data which are asymptotically flat at space-like infinity. The existence of hyperboloidal initial data which do not necessarily satisfy (1.2) will be discussed by two of us (L. A. and P. T. C.) in a subsequent paper [2].

The momentum constraint equations and our geometric assumptions now imply that trg̃(χ̃) = const.(≠ 0 in the case Λ = 0) and the Hamiltonian constraint equation takes, possibly after a rescaling of g̃ by a constant conformal factor, the form R(g̃) = −6. We consider now the tensor fields

Ω, t = \frac{1}{3}D_{i}D^{i}Ω,  (1.3)
s_{ij} = −Ω^{−1}(D_{i}D_{j}Ω − \frac{1}{3}h_{ij}D_{k}D^{k}Ω),  (1.4)
d_{ij} = Ω^{−1}(R_{ij}(h) − \frac{1}{3}h_{ij}R(h) − s_{ij}).  (1.5)

which are derived on M̃ by using Ω, h as well as the covariant derivative operator D and the Ricci tensor defined by h. A discussion of the origin and meaning of these expressions and their generalizations to the case where (1.2) is not imposed may be found in the paper [6] and in the references given therein. Let us note here that the tensor field C_{ij} = Ωd_{ij} is the electric part of the conformal Weyl tensor on M̃ of the corresponding 4-dimensional metric and that it contains all the information about the conformal Weyl tensor on M̃ because the magnetic part of the Weyl tensor vanishes under our hypothesis (1.2). It has been shown by Penrose in [15] that the Weyl tensor vanishes on I where the latter is smooth and admits spherical sections. Therefore we say that the solution (M̃, g̃, χ̃) of the constraint equations for which the relations (1.1), (1.2) are satisfied, is a “smooth hyperboloidal initial data set” if the tensor fields (1.3), (1.4), (1.5) extend to smooth tensor fields on all of M. It may be emphasized here that we do not impose the condition that ∂M be homeomorphic to a union of 2-spheres.

It is our aim to construct general smooth hyperboloidal initial data satisfying (1.2) from suitably chosen “free data”. The discussion above suggests constructing the hyperboloidal initial data as follows. Give as free data an arbitrary 3-dimensional, oriented, compact, smooth Riemannian space (M, h) with boundary ∂M. On M choose a defining function ρ of ∂M, i.e. a smooth function ρ which is positive on M̃ and satisfies on ∂M the conditions ρ = 0, dρ ≠ 0. Find a smooth function u on M̃ such that

R(u^{4}ρ^{−2}h) = −6 and u ≥ u_{0} = const. > 0 on M̃.  (1.6)

Then (M̃, g̃ = u^{4}ρ^{−2}h, χ̃ = \frac{1}{3}trg̃(χ̃)g̃), with suitably defined constant trg̃(χ̃), is a solution of the constraint equations which satisfies (1.1) with Ω = u^{−2}ρ. It qualifies as a smooth hyperboloidal initial data set only if we can show the smooth extensibility of the tensor fields (1.3), (1.4), (1.5).

As will be discussed in detail below the first equation of (1.6), if expressed with respect to the metric h, is an elliptic partial differential equation which degenerates on ∂M. Thus we have to show the existence of a solution of this equation which is smooth and positive up to the boundary on M and on top of this we have to demonstrate the smooth extensibility of the tensor fields (1.3), (1.4), (1.5). This is a highly overdetermined problem which suggest that our data are not so free as assumed. It is not clear a priori which conditions have to be imposed on the data (M, h) such that they lead to smooth hyperboloidal data. The requirements turn out to be surprisingly simple and geometrical.

Theorem 1.1 Suppose (M, h) is a 3-dimensional, orientable, compact, smooth Riemannian space with boundary ∂M. Then there exists a unique solution u of (1.6) and the following conditions are equivalent.

1. The function u as well as the tensor fields (1.3) (1.4) (1.5), determined on M̃ from h and Ω = u^{−2}ρ extend smoothly to all of M.

2. The conformal Weyl tensor C_{ij} = Ωd_{ij} goes to zero at ∂M.

3. The conformal class of h is such that the condition

λ^{∗} = 0 on ∂M  (1.7)

is satisfied, where λ^{∗} denotes the trace-free part of the second fundamental form implied by the metric h on ∂M.

Theorem 1.1 will be proved in §5 as a consequence of Theorems 1.2 and 1.3 discussed below.

The discussion of condition 2 of Theorem 1.1 in §5 shows how the smoothness of the conformal initial data set at ∂M is related under our assumptions to the fall-off behaviour of the conformal Weyl tensor at ∂M. Condition 3 tells us how we have to choose the “free data” (M, h) to construct smooth hyperboloidal data. Apart from our simplifying assumption on the second fundamental form χ̃ on M̃ (which is relaxed in [2]) these data are as general as we can expect. It is a remarkable fact that the analysis of the constraint equation on the hypersurface M together with the smoothness requirements do not yield more stringent conditions than already known from the analysis of the full Einstein equations in a neighbourhood of the conformal boundary I: In the case Λ = 0 condition (1.7) is equivalent to the well known property that I is shear-free on ∂M.

In the case of a cosmological constant Λ < 0 our result opens the possibility to investigate the existence of asymptotically simple solutions to Einstein’s field equations by analysing initial boundary value problems for the regular conformal Einstein equations (cf. [7]) where initial data are prescribed on a hyperboloidal hypersurface and boundary data are given on the time-like conformal boundary I. For an application in the case Λ = 0 the reader is referred also to the article [7].

From the point of view of these applications it is of interest, that the result on the smoothness of the tensor fields (1.3), (1.4), (1.5) is obtained by imposing conditions on the ”free data” only on ∂M. Moreover, the proof of the smoothness requires the discussion of u only in an arbitrary small neighbourhood of ∂M. This suggests that the analysis will generalize in particular to the case where massive sources are present which have spatially compact support in the physical space-time. It is also remarkable that besides the orientability of M, which we assumed, no further conditions on the topology of M enter the discussion.

To avoid misunderstandings it may be remarked that the result of Theorem 1.1 leaves, of course, open the question under which circumstances in the case Λ = 0 smooth hyperboloidal data can arise by Einstein evolution from asymptotically euclidean standard Cauchy data for Einstein’s equations. We hope, however, that our analysis and its subsequent generalization will contribute to the understanding of this question.

We turn now to a discussion of the technical results underlying the proof of Theorem 1.1. The problem of constructing solutions to the constraint equations implied by Einstein’s field equations on space-like hypersurfaces is closely related to the Yamabe problem. In fact, finding a solution u to (1.6) is a Yamabe problem and the first of conditions (1.6), if written as a partial differential equation for the unknown u, is the special case of the Lichnerowicz equation which is known now as the Yamabe equation. In various contexts the investigation of this special case has been observed to be crucial for the understanding of the properties of the solutions to the Lichnerowicz equation in general.

The Yamabe problem in the category of compact Riemannian manifolds has received its solution in the paper [16] by R. Schoen. The Yamabe problem in the case of an open manifold has been studied under various assumptions on the structure of the manifold and on the sign of the Ricci scalar. The following result outlines the situation which we shall analyse in §3 and §4 of this article.

Theorem 1.2 Let (M, h) be a smooth, orientable Riemannian manifold of dimension n ≥ 3 with metric h and compact manifold M with boundary ∂M. Suppose that R̃ is a negative real number. Then there exists a unique smooth function w such that w is positive on the interior M̃ of M, satisfies w(p) → ∞ as p → ∂M, and the metric g̃ obtained on M̃ from h by the conformal rescaling g̃ = w^{4/(n−2)}h has Ricci scalar R̃.

Moreover, if ρ is a smooth defining function for the boundary of M, such that ρ is positive on M̃ while ρ = 0 and |dρ|_{h} = 1 on ∂M, the function w satisfies wρ^{(n−2)/2} → (−n(n − 1)/R̃)^{(n−2)/4} as p → ∂M and thus the space (M̃, g̃) is complete.

A similar result has been obtained by Loewner and Nirenberg [9] under simplifying assumptions on (M, h) and the existence part has been shown in general by Aviles and McOwen [3], [4]. These authors also studied situations which in various respects were more general then the one described above. The uniqueness part of the theorem will be discussed in this paper and the solution will be studied in detail.

It is clear that the rescaled metric would not have the completeness property if the function w would remain bounded along a sufficiently smooth curve which ends on the boundary. We leave it as an open question, whether the requirement of the completeness of g̃ and the condition on its Ricci scalar implies that in fact w goes to infinity at the boundary; such a result has been announced without proof in [12]. The answer to this question is, however, not important for the application we shall study later on.

One of the main results of this paper is a careful analysis of the behaviour of the solution w near ∂M. It is easy to see that by a rescaling of g̃ with a suitable constant conformal factor we can achieve R̃ = −n(n − 1). This value of R̃ will be assumed in the following. For our analysis it will be convenient to consider instead of w the function u = wρ^{(n−2)/2} and to introduce the metric g = ρ^{−2}h on M̃. The function u is then the unique solution, positive and continuous on M, to the boundary value problem

L_{g}u + n(n − 1)u^{(n+2)/(n−2)} = 0  (1.8)
u|_{∂M} = 1  (1.9)

where L_{g} denotes the conformally invariant Laplacian

L_{g} = 4\frac{n − 1}{n − 2}Δ_{g} + R(g)

with Δ_{g} = −tr∇d the Laplace-Beltrami operator and R(g) the Ricci scalar defined by g.

As mentioned before, the Laplacian L_{g} considered as an operator on the compact space (M, h) is degenerate at the boundary. Therefore the classical regularity theory for boundary value problems does not apply. In fact, Δ_{g} is an operator of the kind considered in eg. [11], [13] or [1]. In the papers [11] and [13] the regularity of parametrices of elliptic V_{0}-operators was considered, using a technique of microlocalizing ”near the boundary”, which as a byproduct gives statements about invertibility of Δ_{g} in a setting of weighted Sobolev spaces. In [1] the Fredholm and isomorphism property of certain elliptic systems was considered in a similar setting. In this case, the main tool was the use of the McKean inequality, rather than microlocal techniques. In the present paper we show how to get statements about the regularity of solutions to problems like (1.8) starting from the isomorphism property of the operator rather than from statements about the regularity of the parametrix. The main technical step is to get tangential regularity by commuting V_{b} operators through the equation, see §4. This approach is quite elementary and it appears easier to apply in the case of systems, where the analysis of the parametrix can be rather cumbersome, see [10] for an example.

Recently, the techniques developed by Melrose and Mazzeo have been independently applied to the type of problem considered in the present paper by Mazzeo, see [12]. The theme of that paper is the polyhomogeneity of solutions to the general singular Yamabe problem, i.e. with lower dimensional boundary.

The most important technical results of the present paper are stated in the following theorem. Here the equality in (1.10) is to be understood in terms of polyhomogeneous expansions (cf. Definition 2.5 below, where the notion of tangential smoothness is also discussed). To avoid any ambiguities, let us stress that in our convention M is a compact manifold with its boundary included. Thus, e.g., C^{∞}(M) = C^{∞}(M̄) is a space of functions which are C^{∞} up to the boundary.

Theorem 1.3 On the manifold (M̃, g) let u ∈ C^{0}(M) ∩ C^{∞}(M̃) be the unique positive solution to the boundary value problem (1.8) and (1.9), which exists by Theorem 1.2. Then:

1. u has an asymptotic expansion

u = 1 + ∑_{i=1}^{∞} ∑_{j=0}^{N_{i}} u_{ij}(log ρ)^{j}ρ^{i}  (1.10)

at ∂M with functions u_{ij} ∈ C^{∞}(∂M) and nonnegative integers N_{i} such that N_{i} = 0 for i < n=dimM. Furthermore u is tangentially smooth.

If N_{n} = 0 then in fact N_{i} = 0 for all i and u ∈ C^{∞}(M) .

2. For a given metric h there always exist smooth defining functions ρ as considered above such that R(g) = −n(n − 1) + ρ^{n}R_{n} near ∂M with R_{n} ∈ C^{∞}(M). The solution u ∈ C^{∞}(M) if and only if

R_{n} = 0 on ∂M.  (1.11)

This condition describes the property of the (smooth) conformal class of the metric h which decides on the smoothness of the function u at ∂M.

3. In the case of dimension n = 3 we define on ∂M the function

C(h, ∂M, M) = δ · δ · λ^{∗} + λ^{∗} · R^{∗} − \frac{1}{2}tr(λ)λ^{∗} · λ^{∗}  (1.12)

where λ^{∗} denotes the trace free part of the second fundamental form λ induced by h on ∂M, R^{∗} the orthogonal projection of Ric(h) on ∂M, and the first term on the right hand side of (1.12) is the double divergence of λ^{∗} with respect to the inner covariant derivative on ∂M (cf. §5). This function is a conformal density of weight −3 and

u ∈ C^{∞}(M) if and only if C = 0 on ∂M.  (1.13)

The first two parts of Theorem 1.3 will be proved in §4 and the third part in §5 where we also give an example of a smooth Riemannian manifold (M, h) with boundary for which the quantity C(h, ∂M, M) does not vanish everywhere on ∂M.

2 Preliminaries

2.1 Differential Geometric Considerations

We shall mainly be concerned with Riemannian manifolds of the following type

Definition 2.1 Let (M, h) denote an oriented, compact C^{∞} Riemannian manifold of dimension n ≥ 3 with nonempty boundary ∂M and interior M̃. Assume that ρ ∈ C^{∞}(M) denotes a defining function for ∂M, i.e. ρ > 0 on M̃ while ρ = 0 but dρ ≠ 0 everywhere on ∂M. Then the manifold (M̃, g), where g = ρ^{−2}h, is said to be conformally compact.

Since M is compact with boundary the statement f ∈ C^{∞}(M) means that f as well as all its derivatives on M̃ extend continuously to all of M.

It is easy to see that the Riemannian space (M̃, g) is complete.

The following formulae which express the curvature of g in terms of fields derived from ρ and h will be useful for us.

Riem(g) = ρ^{−2} (Riem(h) − h × (−ρ^{−1}Hess_{h}ρ + \frac{1}{2}ρ^{−2}|dρ|_{h}^{2}h))  (2.1)

where Riem denotes the (4, 0) form of the Riemann tensor and × denotes the Kulkarni-Nomizu product, see [5, 1.110]. It follows from this formula that g has asymptotically negative sectional curvature: K_{g}(p) → −|dρ|_{h}(q) as p → q ∈ ∂M. Taking contractions, we get

Ric(g) = Ric(h) + ρ^{−1}[(n − 2)Hess_{h}ρ − (Δ_{h}ρ)h] − (n − 1)ρ^{−2}|dρ|_{h}^{2}h  (2.2)
R(g) = −n(n − 1)|dρ|_{h}^{2} − (2n − 2)ρ(Δ_{h}ρ) + ρ^{2}R(h)  (2.3)
R̃ic(g) = R̃ic(h) + \frac{n − 2}{ρ} (Hess_{h}ρ + \frac{Δ_{h}ρ}{n}h)  (2.4)

where for any metric k the symbol R̃ic(k) denotes the trace-free part of its Ricci tensor.

Let h be a metric, Θ a positive function, and set h̃ = Θ^{4/(n−2)}h. The conformally covariant Laplacian of h, L_{h} = κΔ_{h} + R(h) with κ = 4(n − 1)/(n − 2), is related to the Ricci scalar of h̃ by

L_{h}Θ = Θ^{(n+2)/(n−2)}R(h̃).  (2.5)

The conformal covariance of the operator L_{h} is expressed by the identity

L_{h}u = Θ^{(n+2)/(n−2)}Lh̃(Θ^{−1}u)  (2.6)

which holds for all sufficiently smooth functions u.

Suppose that (M̃, g) is a conformally compact Riemannian manifold with g = ρ^{−2}h as described in Definition 2.1. Furthermore, assume that u ∈ C^{0}(M) ∩ C^{∞}(M̃) is a positive solution to the boundary value problem (1.8), (1.9). Thus the Ricci scalar of the metric g̃ = u^{4/(n−2)}g takes the value R(g̃) = −n(n − 1). By (2.6) these conditions will remain preserved under transformations h → Θ^{4/(n−2)}h (resp. ρ → Θ^{−2/(n−2)}ρ) and u → Θ^{−1}u, where Θ is a positive smooth function on M with Θ = 1 on ∂M. In what follows we shall exploit this freedom at various occasions without mentioning it each time.

The analysis of the behaviour of the solution u near the boundary is done most conveniently in terms of functions, frame fields, and a choice of conformal scaling which are well adapted to the boundary. Let (M, h, ρ, g) be given as in Definition 2.1. Denote by x the smooth function, defined in an open neighbourhood U of ∂M, which satisfies x = 0 on ∂M, |dx|_{h}^{2} = 1 on U, and for which grad_{h}x is inward pointing on ∂M.

Let x_{0} be a positive number.We denote by Φ the map which sends a point (p, x′) ∈ ∂M × [0, x_{0}[ onto the point q on the integal curve of grad_{h}x through p at which the function x takes the value x′. For x_{0} chosen sufficiently small this map is well defined and in fact a diffeomorphism of ∂M × [0, x_{0}[ onto an open neighbourhood of ∂M. We assume U to coincide with such a neighbourhood. We write Φ^{−1}(q) = (x′, p), consider (x, p) with 0 ≤ x < x_{0}, p ∈ ∂M as ”Gauss coordinates” on U, and call U a ”Gauss domain”. For s ∈ [0, x_{0}[ we set N_{s} = {p ∈ U|x(p) = s}.

Let {e_{a}}_{a=1,..n} denote a smooth orthonormal frame with respect to h, defined near a given point of ∂M, parallely propagated along the integral curves of grad_{h}x, and such that e_{1} = grad_{h}x. Such a frame will be called a ”Gauss frame” in the following. We denote by Γ_{ac}^{b} the connection coefficients with respect to this frame and assume in the following, that tensor indices refer to this frame.

The vector fields e_{A}, A = 2, .., n define a frame on each hypersurface M_{s}. With respect to this frame the coefficients of the second fundamental form λ induced by h on the hypersurface M_{s} are given by λ_{AB} = Γ_{AB}^{1}. The components in the frame {e_{a}} of the covariant derivative of a tensor field T in the direction of e_{1} are obtained by taking the ordinary derivatives of the components of T with respect to the parameter x along the integral curves of grad_{h}x. Thus tensors can be conveniently analysed in terms of power series in x near ∂M. We denote covariant differentiation in the direction of e_{1} by D_{x}.

The Laplace-Beltrami operator defined by h takes on U the form

Δ_{h}u = −D_{x}^{2}u + Δ_{k}u + tr(λ)D_{x}u  (2.7)

where Δ_{k} denotes the Laplace-Beltrami operator defined by the interior metric k induced by h on the slice M_{s} if Δ_{h} is considered at a point with x = s.

Lemma 2.1 Let (M, h) be given as in Definition 2.1. Then

1. A defining function ρ for ∂M can be chosen such that the Ricci scalar of the metric

g = ρ^{−2}h satisfies

R(g) = −n(n − 1) + R_{n}ρ^{n}  (2.8)

with R_{n} ∈ C^{∞}(M).

2. In addition to this we can assume, possibly after rescalings of ρ and h by smooth positive factors (ρ ⟶ Θρ, h ⟶ Θ^{2}h), that

|dρ|_{h} = 1 whence ρ = x near ∂M.  (2.9)

3. With this choice of ρ and scaling of h

R(g) = x^{2}R(h) − 2(n − 1)xtrλ − n(n − 1) near ∂M  (2.10)

from which follows in particular that

trλ|_{∂M} = 0  (2.11)

Proof: By choosing ρ with |dρ|_{h} = 1 on ∂M it follows from (2.3) that R(g) = −n(n − 1) on ∂M. Taking formal derivatives of the same equation and restricting to ∂M gives for k ≥ 1 relations

D_{x}^{k}R(g)|_{∂M} = 2(n − 1) (ρD_{x}^{k+2}ρ − (n − k)D_{x}ρD_{x}^{k+1}ρ) + F_{k}

where F_{k} depends on the functions D_{x}^{j}ρ with j ≤ k and interior derivatives of these on ∂M. Thus we can determine a defining function ρ with derivatives D_{x}^{k}ρ for 1 ≤ k ≤ n on ∂M such that R(g) vanishes there up to order n − 1. For given metric h these derivatives are fixed uniquely and then also D_{x}^{n}R(g)|_{∂M} is determined.

Let Θ be a smooth positive function on M. The metric g remains unchanged if h and ρ are replaced by ĥ = Θ^{2}h and ρ̂ = Θρ respectively. By our choice of ρ we can write 1 − |dρ|_{h} = ρf with a smooth function f. The condition |dρ̂|ĥ = 1 takes thus the form

2Θ < grad_{h}ρ, dΘ > +ρ|dΘ|_{h}^{2} = Θ^{2}f.

By the general theory of first order scalar PDE’s there exists near ∂M a unique smooth solution Θ to this equation such that Θ coincides on ∂M with a given smooth positive function Θ_{0}. This gives the first part of (2.9). From now on we omit hats on ĥ, etc., thus |dρ|_{h} = 1. Define a vector field X by dρ = h_{ij}X^{i}dx^{j}; from (h_{ij}X^{i}X^{j})_{,k} = 0 it follows

X^{i}X^{k}_{;i} = 0 ,

which shows that the integral curves of X are affinely parametrized geodesics (of the metric h) orthogonal to ∂M, and h_{ij}X^{i}X^{j} = 1 implies that for x small enough ρ is the affine parameter along those geodesics, hence ρ = x.

Equation (2.10) follows with this scaling of h and with ρ = x from equation (2.3) and equation (2.7). Equation (2.11) follows by applying D_{x} to equation (2.10) and evaluating on ∂M. □

Remark 2.1 The proof also shows that under the above rescaling of ρ and h with Θ the function R_{n}|_{∂M} scales with a factor Θ_{0}^{−n} ≡ (Θ|_{∂M})^{−n}, whence the property that R_{n}|_{∂M} ≢ 0 is a property of the (smooth) conformal structure of h.

For later use we note the expression of the Laplace-Beltrami operator of g in terms of h and ρ

Δ_{g}u = ρ^{2}Δ_{h}u + (n − 2)ρ < grad_{h}ρ, du > .  (2.12)

2.2 Function Spaces and Operators

Define the set V_{b} to be the set of smooth vector fields on M which are tangent to ∂M and define V_{0} to be the subset of smooth vector fields on M which vanish on ∂M. Let (y^{1}, y^{2}, …, y^{n}) be a coordinate system defined in a neighbourhood of a point of ∂M such that y^{1} = 0 on ∂M and denote y^{1} also by x. Then the V_{b} and the V_{0} vector fields have local expressions

Y = Y^{1}(x∂_{y^{1}}) + Y^{2}∂_{y^{2}} + ⋯ + Y^{n}∂_{y^{n}}

respectively

Y = Y^{1}(x∂_{y^{1}}) + Y^{2}(x∂_{y^{2}}) + ⋯ + Y^{n}(x∂_{y^{n}})

with Y^{i} ∈ C^{∞}. Following Melrose and Mazzeo we introduce the space of operators OP_{b} (considered in [14]) and its subspace OP_{0} (considered in [11] and [13]) generated by vector fields in V_{b} and V_{0}, respectively. These spaces of operators can of course be extended to include corresponding pseudo-differential operators, but we will not need this in the present context.

For given integers m ≥ 1 and β ≥ 0 we denote by OP_{b}^{m,β} the subset of operators of order m in OP_{b}, which in the coordinates considered above have local expression

A = ∑_{|α|≤m} a_{α}x^{α_{1}+β}∂_{y}^{α} , a_{α} ∈ C^{∞} ,

where we have used standard multiindex notation such that e.g. ∂_{y}^{α} = ∂_{y^{1}}^{α_{1}} ⋯ ∂_{y^{n}}^{α_{n}}. We will write OP_{b}^{m} for OP_{b}^{m,0}. Furthermore we denote by OP_{0}^{m} the subspace of operators of order m in OP_{0}, which have local expressions

A = ∑_{|α|≤m} a_{α}x^{|α|}(∂_{y})^{α} , a_{α} ∈ C^{∞} .

Let us recall some definitions of function spaces over M̃. Because of the completeness of the space (M̃, g) and the behaviour of its sectional curvature near ∂M, it is easily seen that there exists a ∈ IR, a > 0 such that for any x ∈ M̃ the set

B̄_{a}(x) = {y ∈ M|d_{g}(x, y) ≤ a}

is a convex normal neighbourhood of x. For tensor fields u, g on M̃ and given points x ∈ M̃, y ∈ B̄_{a}(x) we denote by ũ(x), ũ(y), and g̃ the corresponding tensors at x resp. y given by their components with respect to a normal frame associated with a normal neighbourhood centered at x. Set a(x) ≡ a, and define a space C_{δ}^{s,α}(M, g) as follows:

Definition 2.2 For non-negative integers s and real numbers α, δ with 0 < α < 1, let the weighted Hölder space C_{δ}^{s,α}(M, g) be the space of tensors on M̃ which are locally of C^{s,α} differentiability class, for which the norm

||u||_{C_{δ}^{s,α}(M,g)} = max_{k≤s} ||∇_{k}u||_{C_{δ}^{0,α}(M,g)}  (2.13)

is finite, where

||u||_{C_{δ}^{0,α}(M,g)} = sup_{x∈M̃} ρ^{−δ}|u|_{g} + sup_{x∈M̃} ρ^{−δ} (  sup_{d_{g}(x,y)≤a(x);x≠y} \frac{|ũ(x) − ũ(y)|g̃}{d_{g}(x, y)^{α}} ) . ,  (2.14)

and where ∇_{k}u denotes the covariant differential of order k of u with respect to the covariant derivative ∇ defined by g and |.|_{g} denotes the pointwise tensor norm with respect to g.

Remark 2.2 1. Note that these spaces are the same as the spaces used in [1] but with opposite sign on δ.

2. We denote by C_{δ}^{s}(M, g) the spaces defined by norms similar to those given in (2.13), where, however, the second term on the right of (2.14) is dropped now.

3. In the following we set C_{δ}^{∞} = ⋂_{s=0}^{∞} C_{δ}^{s} and use analogous conventions for other weighted spaces.

We can characterize the Hölder spaces above also by norms which are given in terms of the metric h. For a section u of T_{m}^{r}(M) the tensor norm with respect to g is also given by |u|_{g} = ρ^{m−r}|u|_{h}. To avoid the factors ρ^{m−r} we shall in the following only consider the case m = r = 0. To emphasise the distinction of the covariant derivatives involved we shall denote by D the covariant derivative defined by h. The following weighted Hölder norms are equivalent to the norms (2.13)

||u||_{C_{δ}^{s,α}(M,h)} = max_{k≤s} ||D_{k}u||_{C_{δ−k}^{0,α}(M,h)}  (2.15)

where the norms ||u||_{C_{δ}^{0,α}(M,h)}are defined as in (2.14) with g replaced by h and a(x) = min(rad, d_{h}(x, ∂M))/2, where rad is the injectivity radius of h. In an obvious way we get equivalent norms for the spaces C_{δ}^{s}(M, g). For a review of various properties of the weighted spaces cf. e.g. [8].

Definition 2.3 We denote by C_{δ,b}^{s} the space

C_{δ,b}^{s} = {u ∈ C_{δ}^{s} | D_{Y_{1}} ⋯ D_{Y_{k}}u ∈ C_{δ}^{s−k}, 0 ≤ k ≤ s, Y_{i} ∈ V_{b}}

Definition 2.4 Let (M, h, ρ) be as above. For given real numbers a, α with a < α we denote by W_{α}^{a} (respectively W_{α,b}^{a}) the set of functions f ∈ C^{∞}(M̃) which have the following property: there exist non negative integers N, N_{i}, i = 1, .., N and real numbers s_{0} = a < s_{1} < s_{2} < .. < s_{N} with s_{N} < α and there exists a smooth defining function ρ of ∂M as well as functions f_{ij} ∈ C^{∞}(M) for i = 0, 1, .., N and j = 0, 1, .., N_{i} satisfying

< df_{ij}, grad_{h}ρ >= 0 near ∂M  (2.16)

such that

f = ∑_{i=0}^{N} ∑_{j=0}^{N_{i}} f_{ij}(log ρ)^{j}ρ^{s_{i}} + g  (2.17)

with some function g ∈ C_{α}^{∞} (respectively g ∈ C_{α,b}^{∞}). For given c, a < c ≤ s_{N}, we denote by W_{α,b}^{a,c} the set of functions f ∈ W_{α,b}^{a} for which N_{i} = 0 for s_{i} < c.

Condition (2.16) makes precise the sense in which f_{ij} ∈ C^{∞}(∂M) is to be understood in Theorem 1.3. The size of the neighbourhood of ∂M in which the representation (2.17) holds is unimportant. Also, if for a given f there exists such a representation with a particular defining function, a similar representation can be obtained with respect to any other defining function. In particular we can choose the defining function to coincide on some Gauss domain with the function x.

Definition 2.5 Under the assumptions of Definition 2.4 we say that a function f on M̃ is polyhomogeneous if for some real number a it holds that f ∈ A_{phg}^{a} ≡ ⋂_{α>a} W_{α,b}^{a}. In this case f has an asymptotic expansion at ∂M in the sense of (2.17) for all N in IN, which by a common abuse of notation we write as the equality,

f = ∑_{i=0}^{∞} ∑_{j=0}^{N_{i}} f_{ij}(log ρ)^{j}ρ^{s_{i}} ,  (2.18)

with N_{i}, s_{i}, f_{ij} as in Definition 2.4 (cf. [13]). We denote by A_{phg}^{a,c} the space of f ∈ A_{phg}^{a} such that f has an expansion of the form (2.18) with N_{i} = 0 for all i such that s_{i} < c.

The following result which spells out in particular the tangential regularity of the polyhomogeneous functions is immediate from the definitions.

Proposition 2.2 (Mapping properties) 1. Let A ∈ OP_{b}^{m,β} and let s ≥ m. Then A defines continuous mappings

A : C_{δ,b}^{s,α} → C_{δ+β,b}^{s−m,α} and A : C_{δ}^{s,α} → C_{δ−m+β}^{s−m,α}

2. Under the same assumptions A defines mappings

A : W_{α,b}^{a} → W_{α+β,b}^{a+β} and A : A_{phg}^{a} → A_{phg}^{a+β} .

3. Let A ∈ OP_{0}^{m} and let s ≥ m. Then A defines a continuous mapping

A : C_{δ}^{s,α} → C_{δ}^{s−m,α}

Proposition 2.3 (Commutation Property) Suppose we have operators A ∈ OP_{0}^{m} and B ∈ OP_{b}^{p} with m, p ≥ 1. Then their commutator [A, B] = AB − BA satisfies

[A, B] = ∑_{λ∈I} C_{λ}D_{λ}

where I is a finite set, C_{λ} ∈ OP_{0}^{m}, and either D_{λ} ∈ OP_{b}^{p−1} or D_{λ} = 1, the identity operator (which is always the case if p = 1).

Proof: The general case is readily reduced to the case where A = A_{m}..A_{2}A_{1} with A_{i} ∈ OP_{0}^{1} and B = B_{p}..B_{2}B_{1} with B_{j} ∈ OP_{b}^{1}. The case p = 1 follows by induction on m: It is obvious geometrically and easily checked analytically that [A, B] ∈ OP_{0}^{1} if A ∈ OP_{0}^{1} and B ∈ OP_{b}^{1}. Since

[A_{m+1}A_{m}..A_{1}, B] = A_{m+1}[A_{m}..A_{1}, B] + [A_{m+1}, B]A_{m}..A_{1}

the result follows immediately for all m ≥ 1. The case of arbitrary m, p ≥ 1 follows now by induction on p: having shown it already for p = 1 and using the identity

[A, B_{p+1}...B_{1}] = [A, B_{p+1}]B_{p}...B_{1} + B_{p+1}[A, B_{p}..B_{1}],

the result follows from the induction hypothesis. □

3 Existence and Uniqueness of Solutions

The proof of the existence and the uniqueness of the solution to (1.8), (1.9) is given by a series of lemmas.

Lemma 3.1 Let (M, g, h, ρ) be given as in the Definition 2.1 of conformally compact manifolds, with ρ chosen such that that |dρ|_{h} = 1 on ∂M. Suppose that w is a smooth positive solution on M̃ of

L_{h}w + n(n − 1)w^{(n+2)/(n−2)} = 0 with w → ∞ at ∂M.

Then

lim inf_{p→∂M} (wρ^{(n−2)/2}) (p) ≥ 1

Proof: Because of the conformal covariance of the equation satisfied by w we may assume without loss of generality that the Ricci scalar R(h) is positive near ∂M. In case this condition were not satisfied we could replace h and ρ by Θ^{2}h and Θρ respectively, where the smooth function Θ > 0 is chosen as follows. Fix Gauss coordinates as described in §2 and set Θ = ax^{2} + 1 near ∂M. It follows from the transformation law (2.3) that we can determine the real number a such that R(Θ^{2}h) > 0 near ∂M.

Consider now Gauss coordinates defined on a Gauss domain U on which 0 ≤ x < x_{0} for some positive x_{0} and R(h) ≥ 0. We may assume that ρ = x on U. Choose numbers C, ϵ, ϵ_{1}, x_{1} > 0 with C < 1, x_{1} < x_{0}, ϵ ≤ ϵ_{1}, x_{1} + ϵ_{1} < x_{0}, and consider on the subset W of U where 0 < x ≤ x_{1} the function v = C {(x + ϵ)^{−(n−2)/2} − (x_{1} + ϵ)^{−(n−2)/2}}. The calculation shows that we can fix ϵ_{1}, x_{1} suitably small such that

L_{h}v + n(n − 1)v^{(n+2)/(n−2)} = Cv^{(n+2)/(n−2)} (n(n − 1)(C^{4/(n−2)} − 1) + O(x + ϵ)) ≤ 0

on W for all values of C, ϵ in the range allowed above. Furthermore, we have w > v near the boundary of W. It follows now from the maximum principle, see [9] and [3], that v ≤ w on W. Multiplying this inequality by ρ^{(n−2)/2} and considering first the limit ϵ → 0 and then the limit C → 1 gives the result. □

Lemma 3.2 Under the same assumptions on the function w as in Lemma 3.1 there is a constant C > 0 such that

lim sup_{p→∂M} (wρ^{(n−2)/2}) (p) ≤ C

Proof: Let again U be a Gauss domain and x the corresponding coordinate and ρ = x on U. Assume that the sequence of points p_{k} ∈ M̃ ∩ U is such that

p_{k} → p ∈ ∂M and (wρ^{(n−2)/2}) (p_{k}) → lim sup_{q→∂M} (wρ^{(n−2)/2}) (q).

and set R_{k} = \frac{1}{3}ρ(p_{k}). In the following we assume that k ≥ k_{0} with k_{0} chosen large enough such that the points p_{k} as well as the open balls B_{3R_{k}}(p_{k}) are contained in a convex normal neighbourhood of the point p (in some smooth extension of (M, h)) and that B_{3R_{k}}(p_{k}) ⊂ M̃. If ϕ_{k} ∈ C_{0}^{∞} (B_{3R_{k}}(p_{k})) with ϕ_{k} = 1 on B_{2R_{k}}(p_{k}) then, as has been shown in [4], there is a constant C independent of k, such that

sup_{q∈B_{R_{k}} (p_{k})} w(q) ≤ CR_{k}^{−(n−2)/2} (∫ |∇ϕ_{k}|_{h}^{n}dμ(h) + ∫ ϕ_{k}^{n}dμ(h))^{(n−2)/2n}  (3.1)

where the integration is performed over B_{3R_{k}}(p_{k}). We choose the ϕ_{k} as follows. Let ψ ∈ C^{∞}(IR) be such that ψ(t) = 1 for t ≤ 2,ψ(t) = 0 for t ≥ \frac{5}{2}. Set ϕ_{k}(q) = ψ (\frac{1}{R_{k}} d(q, p_{k})). Then it follows that the second integral on the right of equation (3.1) goes to zero as k → ∞ while the first integrals approach in this limit the values which one would get if one would replace h by a flat metric. The result follows now from the fact that in the flat case the first integral is independent of k. □

Lemma 3.3 Suppose the function w satisfies the same condition as in Lemma 3.1. Assume v is a smooth positive solution on M̃ of L_{h}v + n(n − 1)v^{(n+2)/(n−2)} = 0 such that lim_{p→∂M} (vρ^{(n−2)/2}) (p) = 1.

If either

lim_{p→∂M} (wρ^{(n−2)/2}) (p) = 1

or

ρgrad_{h} (vρ^{(n−2)/2}) is bounded on M̃

then w = v.

Proof: Set h̃ = v^{4/(n−2)}h. Then by equations (2.5) and (2.6) the positive smooth function u = v^{−1}w on M̃ satisfies

Lh̃u = −n(n − 1)u^{(n+2)/(n−2)} = R(h̃)u^{(n+2)/(n−2)}.

and by lemmas 3.1, 3.2 we have

1 ≤ lim inf_{p→∂M} u(p) ≤ lim sup_{p→∂M} u(p) ≤ C  (3.2)

for some constant C ≥ 1 which can be set equal to 1 under our first hypothesis. It follows from the equation for u above that the function z = u − 1 satisfies an equation of the form

Δh̃z + g(u)z = 0

with a function g ∈ C^{1}([0, ∞[) such that g(0) = 0, g(t) > 0 for t > 0. We can now apply the strong maximum principle and the estimates (3.2) to conclude that 0 ≤ z ≤ C − 1 which gives our result under the first hypothesis. In the second case the result follows from Lemma 3.2 and the generalized maximum principle as described e.g.in the article [8]. □

Theorem 3.4 Let (M, g, h, ρ) be given as in the Definition 2.1 of conformally compact manifolds with ρ chosen such that the condition (2.8) is satisfied. Then there exists a solution u to the boundary value problem (1.8), (1.9) such that z = u − 1 ∈ C_{δ}^{∞}(M) for δ < n.

Remark 3.1 Lemma 3.3 implies that this solution to the problem (1.8), (1.9) is unique. In §4 will be seen that the solution u is of class C^{n−1}. This together with Lemma 3.3 and Theorem 3.4 implies Theorem 1.2.

Proof: Assume that h has been scaled such that besides (2.8) also (2.9) is satisfied on some Gauss domain U. Set w = 1 + aρ^{η} with some constants a and 1 ≤ η < n. Then the calculation shows that

L_{g}w + n(n − 1)w^{(n+2)/(n−2)} = 4\frac{n − 1}{n − 2}a(η + 1)(n − η)ρ^{η} + O(ρ^{min(η+1,n)}).

This and the fact that R(h) is bounded on M implies that for a fixed choice of η satisfying the conditions above we can determine sufficiently large positive constants \overline{a}, \underline{a}, b such that the functions

\overline{w} = min(1 + \overline{a}ρ^{η}, b)

and

\underline{w} = min(1 − \underline{a}ρ^{η}, 0)

are weak global upper and lower solutions to our problem. It follows from standard arguments that there exist a solution u with \underline{w} ≤ u ≤ \overline{w} to the boundary value problem (1.8), (1.9) such that u > 0 (cf. [4]) and z = u−1 ∈ C_{η}^{0}(M)∩C^{∞}(M̃). Lemma 3.3 implies that the solution is unique and independent of the choice of η. By elliptic regularity and scaling arguments (cf. e.g. [1]) we conclude from this that z ∈ C_{δ}^{∞} for δ < n. □

This last property is the starting point for our investigation of the smoothness behaviour of the solution near ∂M.

4 Behaviour of the Solution near the Boundary

Using on a suitable Gauss domain U a defining function ρ and a scaling of the metric h as considered in equations (2.8), (2.9), such that ρ = x and observing (2.7), (2.12) we find that the Yamabe equation (1.8), written as equation for the unknown z = u − 1, takes on U the form

A_{x,n}z + Bz = F (p, z)  (4.1)

where we use the notation

A_{x,ξ} = −x^{2}D_{x}^{2} + (n − 2)xD_{x} + ξ(ξ − n + 1)  (4.2)

with ξ = n,

B = x^{2} (Δ_{k} + tr(λ)D_{x})  (4.3)

and

F (p, z) = \frac{n − 2}{4(n − 1)} (−R_{n}x^{n} − R_{n}x^{n}z − n(n − 1)G(z))  (4.4)

with G(z) = (1 + z)^{(n+2)/(n−2)} − 1 − \frac{n+2}{n−2}z. In the following we shall study how the properties of the various terms in equation (4.1) affect the the behaviour of the solution z near ∂M.

4.1 Some ODE Results

In Gauss coordinates the operator (4.2) involves only partial derivatives with respect to x. For fixed ξ ≥ n − 1 consider the equation

A_{x,ξ}u = f,  (4.5)

where now A_{x,ξ} is viewed as an ODE operator on IR^{+}, together with the boundary condition

u(x) → 0 as x → 0_{+}.  (4.6)

In the case f = 0 equation (4.5) is Euler’s equation. Its solutions are given by

u = u_{0}x^{ξ} + u_{−}x^{−ξ+n−1}  (4.7)

with constants u_{0}, u_{−}. This allows to determine the solution of equation(4.5) for more general right hand side by standard methods. Let the kernel H_{ξ}(x, x̃) be given by

H_{ξ}(x, x̃) = C_{ξ} (−x^{ξ}x̃^{n−1−ξ}H(x − x̃) + x^{n−1−ξ}x̃^{ξ}H(x − x̃))

with C_{ξ} = (2ξ + 1 − n)^{−1}, where H(s) is the Heavyside step function. Then, for suitably given f, the function

z(x) = ∫ H_{ξ}(x, x̃)f(x̃)x̃^{−n}dx̃  (4.8)

solves A_{x,ξ}u = f.

Since we are interested only in the behaviour of the solutions near x = 0 we consider spaces of functions on I =]0, 1[ which reflect the radial behaviour of the functions considered in the definitions 2.2, and 2.5. We use obvious analogues to the norms (2.15) to define the spaces C_{α}^{s}(I). We denote by A_{phg}^{a}(I) the functions f ∈ C^{∞}(]0, 1]) which have at x = 0 an asymptotic expansion of the form (2.18) with ρ = x and some constants f_{ij}. Finally we set W_{α}^{a}(I) = A_{phg}^{a}(I) + C_{α}^{∞}(I).

Lemma 4.1 Let ξ ≥ n − 1, α > ξ, suppose that f = O(x^{α}), define

H_{ξ}[f](x) = C_{ξ}x^{ξ} (− ∫_{0}^{x} x̃^{−1−ξ}f(x̃)dx̃ + x^{n−1−2ξ} ∫_{0}^{x} x̃^{ξ−n}f(x̃)dx̃) .  (4.9)

Then H_{ξ}[f] is the unique solution to (4.5), (4.6) which behaves as O(x^{α}) at x = 0. Moreover the operator H_{ξ} given by (4.9) defines a map

H_{ξ} : C_{α}^{∞}(I) → C_{α}^{∞}(I) ,

and for a > α it restricts to mappings

H_{ξ} : A_{phg}^{a}(I) → A_{phg}^{a}(I), H_{ξ} : W_{α}^{a}(I) → W_{α}^{a}(I) .

Proof: For functions f satisfying f = O(x^{α}) with α > β the integral

u_{0} = ∫_{0}^{x_{1}} x̃^{−1−β}f(x̃)

exists, and the results follow by straightforward estimations and/or integrations. □

4.2 The Scalar Laplace–Beltrami Operator.

We generalize slightly the right hand side of equation (4.1) and study now on U ≃ [0, x_{0}[×∂M, where x_{0} > 0, the operator

A_{ξ} = Δ_{g} + ξ(ξ − n + 1) = A_{x,ξ} + B  (4.10)

where the operators on the right hand side are given on U by (4.2) and (4.3) respectively. The interval I in Lemma 4.1 can obviously replaced by the interval [0, x_{0}[ without changing any of the results in an essential way. Since the operator B is in OP_{b}^{2,1} it defines by Proposition 2.2 a continuous mapping B : C_{δ,b}^{∞} → C_{δ+1,b}^{∞}. This means that with respect to the radial regularity we can treat B as a lower order perturbation of A_{x,ξ}. Basic for our discussion are the following two Lemmas:

Lemma 4.2 (Isomorphism property) Let s ≥ 0; for ξ ≥ n − 1 consider the operator

A_{ξ} = Δ_{g} + ξ(ξ − n + 1) .

1. For 0 < α < 1 and for all δ such that

0 < δ < ξ

the map

A_{ξ} : C_{δ}^{s+2,α}(M) → C_{δ}^{s,α}(M)

is an isomorphism.

2. For 0 < α < 1 and for all δ such that

n − 1 − ξ < δ < ξ

there exists x_{∗}, 0 < x_{∗} < x_{0} (depending only upond n, h_{ij}, ξ and δ) such that for every f_{∗} ∈ C^{s+2,α}(N_{x_{∗}}) the map

A_{ξ} : {f ∈ C_{δ}^{s+2,α}(M̃_{x_{∗}})|f(x_{∗}, ·) = f_{∗}(·)} → C_{δ}^{s,α}(M̃_{x_{∗}}) ,

where M̃_{x_{∗}} = {p ∈ M : 0 < x(p) ≤ x_{∗}}, is an isomorphism.

Proof: Point 1) can be shown by standard techniques (cf. e.g. [8]) using the weak barriers given by ϕ_{±} = ± min(1, Cx^{δ}), with a constant C large enough. Point 2) follows from the fact that the functions ϕ_{±} = ±Cx^{δ} are barriers for x ≤ x_{o}(h_{ij}, n, ξ, δ) (for δ ≤ 0 one should use the generalized maximum principle, cf. [8]). □

For our applications the importance of Lemma 4.2 stems from the following:

Lemma 4.3 Let ξ ≥ n − 1, s ≥ 0, 0 < α < 1, suppose that

A_{ξ}u = f,  f ∈ C_{δ}^{s,α},  n − 1 − ξ < δ < ξ,
u ∈ C^{2}(M̃) ∩ C_{σ}^{0,α} , σ > n − 1 − ξ .

Then

u ∈ C_{δ}^{s+2,α} .

Here C^{2}(M̃) only indicates local differentiability, no norm is involved.

Proof: By elliptic regularity and scaling arguments (cf. e.g. [1]) we have u ∈ C_{σ}^{s+2,α}. Let x_{o}(δ) and x_{o}(σ) be given by Lemma 4.2, point 2), set x_{o} = min (x_{o}(δ), x_{o}(σ)), define u_{o} = u|_{M_{xo}}. If σ > δ there is nothing to prove, suppose thus δ > σ. By Lemma 4.2, point 2) there exists ũ ∈ C_{δ}^{s+2,α}(I_{x_{o}}) (recall that I_{x_{o}} = {p ∈ M : 0 < x(p) ≤ x_{o}}) satisfying ũ|_{M_{xo}} = u_{o}, A_{ξ}ũ = f. Since σ < δ we have ũ ∈ C_{σ}^{s+2,α}, by injectivity of A_{ξ} on {u ∈ C_{σ}^{s+2,α}(I_{x_{o}}) : u_{o} = u|_{M_{xo}}} it follows that u|_{I_{xo}} = ũ, thus u ∈ C_{δ}^{s+2,α}(M). □

Proposition 4.4 (V_{b} regularity) Suppose that for some integer m ≥ 1, all integers s ≥ 0 and for some 0 < α < 1 the operator A ∈ OP_{0}^{m} is an isomorphism A : C_{δ}^{s+m,α} → C_{δ}^{s,α} for all δ ∈ I =]c_{0}, c_{1}[, where c_{1} > c_{0} + 1. Fix a δ, δ′ ∈ I such that δ′ > δ and δ′ − c_{0} > 1. If u ∈ C_{δ}^{m,α} satisfies the equation

Au = f

for some f ∈ C_{δ′,b}^{s,α}, then in fact u ∈ C_{δ′,b}^{s,α}. In particular, if f ∈ C_{δ′,b}^{∞} then u ∈ C_{δ′,b}^{∞}.

Proof: From the isomorphism property ensues that u ∈ C_{δ′}^{s+m,α}. If B ∈ OP_{b}^{1} we get from Proposition 2.2 that Bu ∈ C_{δ′−1}^{s+m−1,α}. But by Proposition 2.3 and Proposition 2.2, we obtain

A(Bu) = [A, B]u + Bf ∈ C_{δ′}^{s−1,α},

whence, since A is an isomorphism for the scale of spaces, we get in fact Bu ∈ C_{δ′}^{s+m−1,α}. We can now apply this argument inductively to prove that that u ∈ C_{δ′,b}^{s+m−1,α}. □

The following result provides radial regularity for solutions of equations of the form

(A_{x,ξ} + B)u = f  (4.11)

for f and u with V_{b} regularity. This result is well known^{1}, we include the proof that follows here because it is much simpler than the usual Mellin transform approach.

Lemma 4.5 1. Assume that f ∈ C_{α,b}^{∞}(M) for α > ξ ≥ n − 1. Suppose that u ∈ C_{ξ−ϵ,b}^{∞}(M) with 0 ≤ ϵ < 1 satisfies (4.11) on U. Then u is of the form u = x^{ξ} (∑_{i=0}^{[α−ξ]} u_{i}x^{i} + u_{∗}) where u_{i} ∈ C^{∞}(∂M) and u_{∗} ∈ C_{α−ξ,b}^{∞}, in particular u ∈ W_{α,b}^{ξ}(M).

2. Assume a ≥ 1, 0 ≤ ϵ < 1 and let u ∈ C_{a−ϵ,b}^{∞} satisfy the equation (4.11) on some neighbourhood of ∂M. If f ∈ W_{α,b}^{a} with α > ξ ≥ n − 1 then u ∈ W_{α,b}^{min(a,ξ)}. In particular, if u satisfies the same conditions as above and f ∈ A_{phg}^{a} for some a ≥ 1 then u ∈ A_{phg}^{min(a,ξ)}.

Proof:

1We are grateful to G. Łysik and B. Ziemian for pointing out to us the Mellin transform approach to this problem.

1. Since B ∈ OP_{b}^{2,1} it follows from Proposition 2.2 that

A_{x,ξ}u = f − Bu ∈ C_{α′,b}^{∞}  (4.12)

where α′ = min(ξ − ϵ + 1, α) > ξ. On a fixed integral curve of the vector field grad_{h}x on U it follows from the ODE (4.12) and from u = o(x^{n−1−ξ}) that u = x^{ξ}u_{0}+H_{ξ}[f −Bu]. Here u_{0} is constant on the integral curves but must be considered as a function on ∂M. It follows from Lemma 4.1 that H_{ξ}[f − Bu] ∈ C_{α′}^{∞}. From the expression for H_{ξ} given in (4.9) it follows that in fact H_{ξ}[f − Bu] ∈ C_{α′,b}^{∞}. Using the smoothness assumptions on u we conclude that u_{0} ∈ C^{∞}(∂M). If α′ < α we can repeat the procedure. Assume that u has been shown to have an expansion of the form

u = x^{ξ} (∑_{i=0}^{l} v_{i}x^{i} + v_{∗})

with v_{i} ∈ C^{∞}(∂M),v_{∗} ∈ C_{α−ξ,b}^{∞}, and l < [α − ξ]. Then we may again use Lemma 4.1 to conclude that

u = x^{ξ}u′_{0} + ∑_{i=0}^{l} v_{i}H_{ξ}[B(x^{i+ξ})] + H_{ξ}[f − B(x^{ξ}v_{∗})]

and it is easily seen that this pushes the expansion one step further while preserving all the required smoothness properties.

2. Let

f = ∑_{i=0}^{I} ∑_{j=0}^{N_{i}} f_{i,j}x^{s_{i}} log^{j} x + r , r ∈ C_{α,b}^{∞} .

It is easily seen by induction that there exist functions v_{i,j} ∈ C^{∞}(∂M), i = 1, …, Ĩ, j = 0, …, N_{i} + 1, for some Ĩ, such that

A_{ξ}(∑_{i=0}^{Ĩ} ∑_{j=0}^{N_{i}+1} v_{i,j}xs̃_{i} log^{j} x) = ∑_{i=0}^{I} ∑_{j=0}^{N_{i}} f_{i,j}x^{s_{i}} log^{j} x + r̃ ,

where {s̃^{i}} ≡ {s_{i} + k : k ∈ IN ∪ {0}, s_{i} + k ≤ α + 1}, for some r̃ ∈ A_{phg}^{α+1} ⊂ C_{α+1−ϵ,b}^{∞} ⊂ C_{ξ−ϵ}^{∞}, for any ϵ > 0. By Lemma 4.2 there exists ũ ∈ C_{ξ−ϵ}^{∞} such that A_{ξ}ũ = r − r̃, by Lemma 4.2 and Proposition 4.4 we have ũ ∈ C_{ξ−ϵ,b}^{∞}. Since both u and ũ + ∑Ĩ ; i=0 ∑_{j=0}^{N_{i}+1} v_{i,j}xs̃_{i} log^{j} x are in C_{a−ϵ}^{∞}, the uniqueness part of Lemma 4.2, point 1, shows that u = ũ+ ∑Ĩ ; i=0 ∑_{j=0}^{N_{i}+1} v_{i,j}xs̃_{i} log^{j} x, and since r −r̃ ∈ C_{α,b}^{∞} we have ũ ∈ W_{α,b}^{ξ} by point 1 of this Lemma.

□

We next consider the regularity of solutions on a conformally compact manifold M to equations of the type A_{ξ}u = f, under various conditions on f:

Theorem 4.6 (Regularity) Let ξ > n−1 and assume that u ∈ C_{δ}^{2}, with some δ > n−ξ, is a solution to the equation

A_{ξ}u = f.  (4.13)

1. If f ∈ C_{α,b}^{∞}(M) for some α > ξ, then u is of the form u = x^{ξ}(∑_{i=0}^{[α−ξ]} u_{i}x^{i} + u_{∗}) where u_{i} ∈ C^{∞}(∂M) and u_{∗} ∈ C_{α−ξ,b}^{∞}.

2. Assume^{2} that a > n − ξ. If f ∈ W_{α,b}^{a} with α > ξ, then u ∈ W_{α,b}^{min(ξ,a)}. In particular, if f ∈ A_{phg}^{a}, then u ∈ A_{phg}^{min(ξ,a)}. If f ∈ A_{phg}^{ξ+ϵ,ℓ} for some ℓ > ξ + ϵ, then u has no log-terms until the order ℓ.

3. Assume that ξ is an integer and that f ∈ C^{∞}(M). Then there exists u_{1}, u_{2} ∈ C^{∞}(M), with u_{2} = O(x^{ξ}), such that

u = u_{1} + u_{2} log x .

If f|_{∂M} = 0, then u|_{∂M} = 0. Further, if in addition f = o(x^{ξ}), then u ∈ C^{∞}(M).

Proof: By Lemma 4.2 and Proposition 4.4 u is tangentially smooth, and point 1 together with the first part of point 2 follow from Lemma 4.5. The second part of point 2 follows by comparing the coefficients in an asymptotic expansion of A_{ξ}u with those of f. To prove point 3, note that by point 2 u is polyhomogeneous and again comparing the coefficients in an asymptotic expansion of A_{ξ}u with those of f one finds that u has an asymptotic expansion of the form

u = ∑_{i=0}^{∞} u_{i,1}x^{i} + log x ∑_{i=0}^{∞} u_{i,2}x^{ξ+i} .

By Borel’s Lemma there exist functions ũ_{1}, u_{2} ∈ C^{∞}(M) with the asymptotic expansions

ũ_{1} = ∑_{i=0}^{∞} u_{i,1}x^{i} , u_{2} = ∑_{i=0}^{∞} u_{i,2}x^{ξ+i},

so that ũ ≡ u − ũ_{1} − u_{2} log x is polyhomogeneous, vanishes to arbitrary order at ∂M together with all derivatives, and thus ũ ∈ C_{∞}(M). It follows that Lũ ∈ C^{∞}(M), and Lũ vanishes to arbitrary order at ∂M, so that point 1 implies ũ ∈ C^{∞}(M). Setting u_{1} = ũ + ũ_{1} we have u = u_{1} + u_{2} log x, and the result follows. □

4.3 Radial regularity for the nonlinear case and proof of Theorem 1.3

In this section we consider solutions to an equation of the form

A_{ξ}w = F (w) ≡ Uw + V + G(w)  (4.14)

where U, V ∈ A_{pgh}^{1} and G ∈ C^{∞}(IR), with G(0) = G′(0) = 0. First we prove that a solution which decays sufficiently rapidly at ∂M must be tangentially smooth.

2The hypotheses a, δ > n − ξ can be relaxed to a, δ > n − ξ − 1 by a difference quotient argument (which is sharp), cf. [2]; this will however not be needed in this paper.

Lemma 4.7 Let ξ > n − 1, δ > max(0, n − ξ), assume that w ∈ C_{δ}^{∞} is a solution of (4.14). Then for all α < ξ we have in fact w ∈ W_{α,b}^{1}, i.e. w is tangentially smooth.

Proof: Suppose that δ < 1/2, then F (w) ∈ C_{2δ}^{∞}, by Lemma 4.2 it follows that w ∈ C_{2δ}^{∞}, and repeating this argument a finite number of times if necessary we get w ∈ C_{δ}^{∞}, δ > 1/2. It then follows

F (w) = F_{η} + R_{η}
≡ ∑_{i=1}^{k} ∑_{j=0}^{N_{i}} F_{i,j}x^{s_{i}} log^{j} x + R_{η} ,  (4.15)

with R_{η} ∈ C_{η}^{∞}, η ≥ min(2δ, 1+δ), s_{k} ≥ η, and F_{i,j} ∈ C^{∞}(∂M). Lemma 4.2 and Theorem 4.6 show that we can find w_{η} ∈ A_{phg}^{1}, r_{η} ∈ C_{η}^{∞} if η < ξ or r_{η} ∈ C_{α}^{∞} for any α < ξ if η ≥ ξ, such that

A_{ξ}w_{η} = F_{η}, A_{ξ}r_{η} = R_{η} .

Uniqueness implies w = w_{η} + r_{η}, and repeating the above argument a finite number of times one obtains η ≥ α, thus w = w_{α} + r_{α}, w_{α} ∈ A_{phg}^{1}, r_{α} ∈ C_{α}^{∞}. Let now B ∈ OP_{b}^{1}, we have

A_{ξ}(Bw) = [A_{ξ}, B]w − BF (w) ∈ W_{α}^{1} ,

and since Bw = Bw_{α} + Br_{α}, Bw_{α} ∈ A_{phg}^{1}, Br_{α} ∈ C_{α−1}^{∞}, α − 1 > 0, one concludes from Lemma 4.2 that Bw ∈ W_{α}^{1}. An induction argument shows that for every B_{i} ∈ OP_{b}^{1}, i = 1, …, N, we have B_{1} ⋯ B_{i}w ∈ W_{α}^{1}, thus w ∈ W_{α,b}^{1}, which had to be established. □

This result enables us to prove polyhomogeneity.

Theorem 4.8 Under the assumptions of Lemma 4.7, we have w ∈ A_{phg}^{1}.

Proof: By Lemma 4.7, w ∈ W_{η,b}^{1} for some η > 1. It follows that

A_{ξ}w = Uw + V + G(w) ∈ W_{η′,b}^{1} with any η′ < η + 1

Applying Theorem 4.6 inductively now shows that w ∈ W_{α,b}^{1} for any α and hence w ∈ A_{phg}^{1}. □

If the term V satisfies an additional requirement, then we can prove that the solution is actually smooth.

Corollary 4.9 If U, V ∈ C^{∞}(M), U = O(x), V = O(x^{ξ+1}), then x^{−ξ}w ∈ C^{∞}(M). In particular, if ξ is an integer, then w ∈ C^{∞}(M).

Proof: By Theorem 4.8, w ∈ A_{phg}^{1} and hence after iterating

A_{ξ}w = Uw + V + G(w)

using Theorem 4.6 part(2) we find that w ∈ A_{phg}^{ξ,2ξ}. Iterating this leads to w ∈ A_{phg}^{ξ,ℓ} for any ℓ and hence the corollary is proved. □

From the results above we can deduce Theorem 1.3. We know already that the Yamabe equation can be written in the form (4.1) with (4.2), (4.3), (4.4) for an unknown z, which by Theorem 3.4 is in C_{δ}^{∞} for δ < n. Since the function G on the right hand side of (4.4) is smooth in a domain strictly larger than the set {z > −1} where z takes its values, we can assume G to be extended suitably to a function that satisfies the requirements on the function G considered above. Now parts 1 and 2 of Theorem 1.3 follow from Lemma 2.1, Theorem 4.8, and Corollary 4.9. Part 3 will follow from parts 1 and 2 and Lemma 5.2, which will be given in the next Section.

5 Smoothness of the hyperboloidal initial data

We shall in the following discuss a number of equivalent conditions under which the solution u of the boundary value problem (1.8), (1.9) extends smoothly to ∂M. These will allow us to prove part 3 of Theorem 1.3 and Theorem 1.1.

We assume that on some Gauss domain U the conditions (2.8) and (2.9) are satisfied. Then we have

R_{n}x^{n} = x^{2}R(h) − 2(n − 1)xtr(λ) on U  (5.1)

Taking x–derivatives of (5.1) and evaluating on ∂M we get

D_{x}^{k}tr(λ) = \frac{k}{2(n − 1)}D_{x}^{k−1}R(h) on ∂M for k = 0, …, n − 2  (5.2)

and the equivalent “regularity conditions”

R_{n}|_{∂M} = 0 ⇔ 0 = D_{x}^{n−1}tr(λ) − \frac{1}{2}D_{x}^{n−2}R(h) on ∂M.  (5.3)

One would like to give the regularity condition in terms of well understood quantities on ∂M which do not involve non-tangential derivatives. This is the more desirable since the quantities entering the expression above depend very much on our choice of scaling. As we have seen in the proof of Lemma 2.1 there is still some freedom in the scaling of h. Changing this scaling will in general also change the family of hypersurfaces M_{s} in U and the second fundamental forms obtained for different scalings will be related to each other in a complicated way.

In the case of dimension n = 3 we can derive a simple condition. First we introduce a useful function.

Lemma 5.1 Let λ_{AB}^{∗} be the traceless part of λ_{AB}, define on ∂M the function

C(h, ∂M, M) = δ^{A}δ^{B}λ_{AB}^{∗} + λ^{∗AB}R_{AB} − \frac{1}{2}tr(λ)λ^{∗AB}λ_{AB}^{∗}.

Then C(h, ∂M, M) is a conformal density of weight −3, i.e. it satisfies C(Θ^{2}h, ∂M, M) = Θ^{−3}C(h, ∂M, M) for all smooth positive functions Θ on M.

Proof: The proof of the transformation behaviour of C under transitions h → Θ^{2}h follows by a straightforward calculation from (2.2) and the transformation law λ → Θλ + k < n, dΘ > for the second fundamental form on ∂M, where n denotes the inward pointing unit normal of ∂M with respect to the metric h. □

This allows us to give the regularity condition in the form

Lemma 5.2 In the case of dimension 3 the solution u to the Yamabe problem (1.8), (1.9) is smooth on M if and only if the conformal structure defined by the metric h is such that C(h, ∂M, M) = 0 on ∂M.

Proof: Using the same tetrad conventions as in Section 2.1, the embedding equations for the hypersurfaces M_{s} into M are given by Gauss’ equation

R_{ABCD} = r_{ABCD} + λ_{AD}λ_{BC} − λ_{AC}λ_{BD}  (5.4)

and its contractions

R_{BD} − R_{1B1D} = r_{BD} + λ_{CD}λ_{B}^{C} − λ_{C}^{C}λ_{BD}  (5.5)
R − 2R_{11} = r + λ_{CD}λ^{CD} − (λ_{C}^{C})^{2}  (5.6)

and by Codazzi’s equation and its contraction

R_{1BCD} = δ_{C}λ_{DB} − δ_{D}λ_{CB}  (5.7)
R_{1C} = δ_{C}λ_{D}^{D} − δ_{D}λ_{C}^{D}.  (5.8)

Furthermore we have the equations

R_{1A1B} = e_{1} (λ_{AB}) − λ_{AC}λ_{B}^{C}  (5.9)
R_{11} = e_{1} (λ_{A}^{A}) − λ_{AB}λ^{AB}.  (5.10)

The curvature quantities indicated by a capital R on the left hand side of these formulae are the Riemann tensor, the Ricci tensor, respectively the Ricci scalar defined by the metric h while on the right hand side are denoted by a small r the corresponding quantities derived from the interior metric k on the hypersurface M_{s} in U. Contractions on the right hand sides are performed with the metric k.

We can replace the expression on the right hand side of (5.3) by a lower order expression in the following way. We take derivatives of (5.10) to obtain

D_{x}^{n−1}tr(λ) = D_{x}^{n−2}R_{11} + D_{x}^{n−2}(λ_{AB}λ^{AB})

From the contracted Bianchi identity and (5.8) we derive

\frac{1}{2}D_{x}R = D^{a}R_{a1} = D_{x}R_{11} − Δ_{k}tr(λ) − δ^{A}δ^{B}λ_{AB} − tr(λ)R_{11} + λ^{AB}R_{AB}.

Taking derivatives of this equation and using the previous equation we arrive at

D_{x}^{n−1}tr(λ) − \frac{1}{2}D_{x}^{n−2}R(h) =

D_{x}^{n−2}(λ_{AB}λ^{AB}) + D_{x}^{n−3} (Δ_{k}tr(λ) + δ^{A}δ^{B}λ_{AB} + tr(λ)R_{11} − λ^{AB}R_{AB})

We assume now n = 3, use equations (5.9), (5.5) to obtain D_{x}λ_{AB} = R_{AB} − r_{AB} + tr(λ)λ_{AB}, contract this with λ^{AB}, and use the resulting expression to replace the first term on the right of the previous equation. We obtain

D_{x}^{2}tr(λ) − \frac{1}{2}D_{x}R(h) =

−2rtr(λ) + 2tr(λ)λ_{AB}λ^{AB} + Δ_{k}tr(λ) + δ^{A}δ^{B}λ_{AB} + tr(λ)R_{11} + λ^{AB}R_{AB}

Observing now the scaling behaviour of C and the fact that in our present scaling condition (2.11) is satisfied, we get our result from (5.3). □

Proceeding in a similar way in the case of higher dimensions would give complicated terms, non linear in the curvature of h.

The following example shows that the condition given in Lemma 5.2 is not trivial. Let M be the solid torus embedded in IR^{3} whose boundary is given by the 2-surface

∂M = {(x, y, z) ∈ IR^{3}|x = (R + r cos θ) cos ϕ, y = (R + r cos θ) sin ϕ, z = r sin θ)}

where 0 ≤ θ, ϕ < 2π and r, R are given numbers satisfying 0 < r < R. Assume that h is the metric induced on M by the flat standard metric on IR^{3}. Then we find

C(h, ∂M, M) = \frac{1}{4}R\{r(R + r cos θ)\}^{−3}(R^{2} + 2r^{2} + 4Rr cos θ)

which vanishes only on sets of measure zero on ∂M.

5.1 Proof of Theorem 1.1

Suppose u is the solution to the boundary value problem (1.8), (1.9), set Ω = ρu^{−2} and g̃ = Ω^{−2}h. We shall study the behaviour of the tensor fields (1.3), (1.4), and (1.5) on a Gauss domain U where we assume the conditions (2.8) and (2.9) to be satisfied. It follows then from Theorem 1.3 that Ω is near ∂M of the form

Ω = x − 2u_{3,0}x^{4} − 2u_{3,1}(log x)x^{4} + higher order.  (5.11)

Thus Ω ∈ C^{3}(M).

The condition that Ω be smooth is equivalent to the condition that u be smooth which imposes the condition C = 0 on our ”free data” (M, h).

Using the above expression for Ω we can calculate from it the trace free part of the Hessian of Ω and get

D_{a}D_{b}Ω − \frac{1}{3}h_{ab}D_{c}D^{c}Ω = −λ_{ab} + \frac{1}{3}h_{ab}tr(λ) + O(x^{2} log x)

where λ is considered as a tensorfield on U which contracts to zero with the gradient of x. Thus the requirement that the field (1.4) extends smoothly to ∂M implies in view of our assumption (2.11) the conformally invariant condition

λ^{∗} = 0 on ∂M  (5.12)

which implies in turn the smoothness of Ω by Lemma 5.2.

Assuming (5.12) to be satisfied and using the equations (5.2), (5.6), (5.9), and (5.10) we find that the field s_{ab} extends smoothly to ∂M and takes there the following values

s_{1A} = 0, s_{11} = −\frac{1}{3}D_{x}tr(λ) = −\frac{1}{12}R, s_{AB} = D_{x}λ_{AB} − \frac{1}{3}h_{AB}D_{x}tr(λ) = R_{AB} − \frac{1}{3}Rh_{AB}.

Let C_{ab} denote the electric part of the Weyl curvature (note that the magnetic part of the Weyl tensor vanishes because of our hypothesis χ̃ = \frac{1}{3}tr(χ̃)g̃); it is now straightforward to check that

C_{ab} = Ωd_{ab} = R_{ab}(h) − \frac{1}{3}R(h)h_{ab} − s_{ab}

goes to zero at ∂M which implies that d_{ab} also extends smoothly to ∂M. Also note that from what has been said it follows that C_{ab} goes to zero at ∂M if and only if s_{ab} is bounded which we have seen to be true if and only if (5.12) is satisfied. From this follows Theorem 1.1.

We end the article with a few remarks. Comparing (1.5) with (2.4) shows that

C_{ab} = R_{ab}(g̃) − \frac{1}{3}g̃_{ab}R(g̃).  (5.13)

Beside the implication of (5.13) for the fall–off behaviour of the physical field we gather from (5.13) that the data are trivial, i.e. determine a conformally flat solution of Einstein’s equations, only if (M̃, g̃) is a space of negative constant curvature, i.e. only if (M, h) is conformally flat. It follows that there exist non-trivial solutions to Einstein’s equations satisfying (5.12). It may be remarked, however, that the conformal flatness of (M, h) does not necessarily imply triviality of the solution. Consider the case where M = {x ∈ IR^{3}|r_{1} ≤ |x| ≤ r_{2}} with real numbers 0 < r_{1} < r_{2} and where h is the metric induced on M by the flat standard metric on IR^{3}. It is easy to see that the regularity condition (5.12) is satisfied. The resulting space (M̃, g̃) cannot be of constant negative curvature since it is complete and simply connected but M̃ is not diffeomorphic to IR^{3}. Furthermore, it follows that the hyperboloidal data and consequently the solution of Einstein’s equations (M, γ) determined from (M, h) are spherically symmetric. By Birkhoff’s Theorem (M, γ) represents thus a portion of a Schwarzschild–Kruskal spacetime with positive mass. We also see from this example that in the Schwarzschild-Kruskal space-time there exist smooth space-like hypersurfaces which extend from one future null infinity to the other future null infinity and on which the second fundamental form is proportional to the interior metric.

Acknowledgements: H.F. should like to thank the I.H.E.S., Paris, and the Royal Institute of Technology, Stockholm, for hospitality. P.T.C. acknowledges a collaboration with J. Jezierski at an early stage of work on this paper; he also wishes to thank J. Jezierski, G. Łysik, R. Mazzeo and B. Ziemian for useful discussions.

References

[1] L. Andersson. Elliptic systems on manifolds with asymptotically negative curvature. Submitted to Indiana Univ. Math. J., 1990.

[2] L. Andersson, P.T. Chruściel, in preparation.

[3] P. Aviles and R.C. McOwen. Complete conformal metrics with negative scalar curvature in compact Riemannian manifolds. Duke Math. J., 56, 395–398 (1988).

[4] P. Aviles and R.C. McOwen. Conformal deformation to constant negative scalar curvature on noncompact Riemannian manifolds. J. Differential Geom., 27, 225– 239 (1988).

[5] A.L. Besse. Einstein Manifolds, volume 10 of Ergebnisse der Math. 3. Folge. Springer, Berlin, 1987.

[6] H. Friedrich. On static and radiative spacetimes. Comm. Math. Phys., 119, 51–73 (1988).

[7] H. Friedrich. On the global existence and the asymptotic behaviour of solutions to the Einstein-Maxwell-Yang-Mills equation. J. Differential Geom., 34, 275–345 (1991).

[8] C.R. Graham and J.M. Lee. Einstein metrics with prescribed conformal infinity on the ball, Adv. Math., 87, 186-225 (1991).

[9] C. Loewner and L. Nirenberg. Partial Differential Equations Invariant under Conformal or Projective Transformations In: Contributions to Analysis, L.V.Ahlfors et al., eds., Academic Press, New York, 1974.

[10] R. Mazzeo. Hodge cohomology of negatively curved manifolds. Technical report, 1986.

[11] R. Mazzeo. The Hodge cohomology of conformally compact metrics. J. Differential Geom., 28, 309–339 (1988).

[12] R. Mazzeo. Regularity for the singular Yamabe problem. Stanford University preprint, 1990.

[13] R. Mazzeo and R.B. Melrose. Meromorphic extension of the resolvent on complete spaces with asymptotically negative curvature. J. Funct. Anal., 75, 260–310 (1987).

[14] R.B. Melrose. Transformation of boundary value problems. Acta Math., 147, 149– 236 (1981).

[15] R. Penrose. Zero rest-mass fields including gravitation: asymptotic behaviour. Proc. Roy. Soc. Lond., A284, 159–203 (1965).

[16] R.M. Schoen. Conformal deformation of a Riemannian metric to constant scalar curvature. J. Differential Geom., 20, 479–495 (1984).