Geometrically finite Poincaré-Einstein metrics
Eric Bahuaud Frédéric Rochon






We construct new examples of Einstein metrics by perturbing the conformal infinity of geometrically finite hyperbolic metrics and by applying the inverse function theorem in suitable weighted Hölder spaces.



Department of Mathematics, Seattle University
bahuaude(at)seattleu.edu

Département des mathématiques, UQÀM
rochon.frederic(at)uqam.ca

[2010]53C21; 53C25, 58J05, 35J57, 35J70



Geometrically finite Poincaré-Einstein metrics
    Frédéric Rochon
    December 30, 2023
==============================================



§ INTRODUCTION


Let  be a smooth compact n-manifold with boundary ∂ M and interior M.  Let ρ∈ C^∞() be a non-negative function vanishing to first order precisely on ∂ M.  A Riemannian metric g on M is C^k,α conformally compact if ρ^2 g extends to a C^k,α metric on .  The induced conformal class of metrics on ∂ M, [(ρ^2 g)|_T∂ M] is the conformal infinity of g.


The Poincaré metric on the unit ball ^n := ^n = {x∈^n: |x| < 1} given by h=4/(1-|x|^2)^2∑ (dx^i)^2 is an example of a conformally compact Einstein metric.  Here ρ = (1-|x|^2)/2, and the conformal infinity is the conformal class of the round metric ĥ on ^n-1. In 1991, via the inverse function theorem, Robin Graham and John M. Lee proved the existence of asymptotically hyperbolic Einstein metrics on the unit ball with prescribed conformal infinity sufficiently close to the Poincaré model <cit.>.  
Such metrics are variously called Poincaré-Einstein, or asymptotically hyperbolic Einstein or conformally compact Einstein metrics.  


This result was subsequently generalized by Biquard <cit.> in complex, quarternionic and octonionic hyperbolic spaces through deformations of the appropriate notion of conformal infinity.  In another direction, Lee <cit.> extended the perturbation result of <cit.> by deforming  the conformal infinity of more general Poincar-Einstein metrics.  To be able to apply the inverse function theorem, Lee needed the L^2-kernel of the linearized problem to vanish, a condition automatically satisfied when the Poincar-Einstein metric has nonpositive sectional curvature, for instance when it is a convex co-compact hyperbolic metric.     More recently, Albin, in <cit.>, generalizes <cit.>, still by deforming the conformal infinity of the hyperbolic space, but replacing the Einstein equation by an equation involving a linear combination of Lovelock tensors, obtaining in this way examples of Poincar-Lovelock metrics.  In a very different direction, let us point out also that Enciso and Kamran in <cit.> have obtained a Lorentzian version of <cit.>.   


More generally, instead of convex co-compact hyperbolic metrics, one can consider  geometrically finite hyperbolic metrics.  As observed by Mazzeo and Philips <cit.>, these metrics admit a natural compactification to a manifold with corners with at most codimension 2 corners.  When the volume is infinite, there is a special boundary hypersurface,  H_0, corresponding to the directions of maximal volume growth, while the other boundary hypersurfaces are mutually disjoint and correspond to cusps.  A cusp of maximal rank yields an end of finite volume and the corresponding boundary hypersurface H_i is a closed submanifold disjoint from all the other boundary hypersurfaces.  If instead the cusp is of intermediate rank, then the corresponding boundary hypersurface H_i is a manifold with boundary with non-empty intersection with H_0, so that H_i = H_i∩ H_0.  When the metric is convex co-compact, H_0 is the only boundary hypersurface and we recover the usual compactification to a manifold with boundary.  


Coming back to a general geometrically finite hyperbolic metric g, this suggests that if ρ now denotes a boundary defining function of H_0, then 
ρ^2g|_TH_0 should be a representative of the conformal infinity of g.  As explained in  <ref>-<ref>, the representative ρ^2g|_TH_0 is naturally a complete metric on H_0∖ H_0 with foliated cusps at infinity in the sense of <cit.>.   Deforming the conformal class of ρ^2g|_TH_0, one could therefore hope to obtain a corresponding Einstein deformation of the metric g, at least for small enough deformations. To avoid certain complications near the corners, one can even restrict to deformations of ρ^2g|_TH_0 compactly supported in an open set U of H_0 with closure disjoint form H_0, since near such an open set, the metric g locally takes the form of a conformally compact metric.  The main purpose of the present paper is to establish that indeed, such Einstein deformations are possible, yielding a new class of complete Einstein metrics that we call  geometrically finite Poincar-Einstein metrics.  The precise result is as follows (see Theorem <ref> below for a statement with more details on the regularity of the metric).     


 
For n ≥ 4, and k > n, let (M^n,h) be a geometrically finite hyperbolic metric with n_c intermediate rank cusps and no cusps of maximal rank.  If n=4, suppose moreover that each cusp is of rank 1.  Then there exists > 0 such that for any smooth compactly supported symmetric 2-tensor ∈ C^∞_c(U;T^*U⊗ T^*U) and perturbation ĝ = ĥ + with _C^k,α(U) <, there is a geometrically finite asymptotically hyperbolic metric with cusps g on M such that 
ρ^2 g is continuous on U with (ρ^2 g)|_TU = ĝ and  g is Einstein, i.e. 

    (g) + (n-1)g = 0.






 We now discuss the key aspects of the proof in a high-level manner.  As in <cit.>, we replace the Einstein equation above by an appropriate gauge-adjusted equation.  In particular we define an operator Q that takes pairs of metrics (g,t) to symmetric two-tensors by

    Q(g,t) = (g) + (n-1)g - δ_g^* (g t^-1 (δ_g (G_g t))),

where (G_g t)_ij := t_ij - 1/2 g^pq t_pq g_ij, δ_g is the divergence of a symmetric 2-tensor and δ_g^* is its formal adjoint.  That it is enough to solve Q(g,t) = 0 for sufficiently regular g and t will follow from the maximum principle.  


It will be important to obtain approximate solutions to Q(g,t) = 0.  This is the step where our assumption that the conformal deformation is compactly supported yields important simplifications.  Indeed, the argument of Graham-Lee to construct an approximate solution is local in the directions tangent to the boundary, so can be applied directly to our setting as long as the conformal perturbations are compactly supported.  


Having obtained a sufficiently good asymptotic solution, the result follows by applying the inverse function theorem.  This requires an isomorphism theorem for an appropriate linear problem related to the linearization of Q at the hyperbolic metric.  Writing r = uh + r_0 where u is a smooth function and r_0 is trace-free with respect to h, we will see the existence problem for a perturbative gauge-adjusted Einstein metric reduces to the invertibility on weighted tensor field spaces of the operator

    L r = 1/2 ( (Δ^h + 2(n-1)) (uh) + (Δ^h - 2) r_0 ).

Comparing to <cit.>, this is the step where a new approach is required.  This starts with the definition of the appropriate weighted Hlder spaces, where, instead of patching together descriptions in local coordinate charts, we were led to take the more global point of view of Melrose <cit.> and Ammann-Lauter-Nistor <cit.> relying on the compactification of Mazzeo-Phillips <cit.>.  


Given these weighted Hlder spaces, the isomorphism theorem is proved using a suitable exhaustion of the geometrically finite hyperbolic manifold by bounded domains with smooth boundaries.  On these domains, we rely on the argument of Koiso as explained in <cit.> to show that the linearized problem with Dirichlet boundary conditions is an isomorphism.  To obtain an isomorphism for the whole manifold, one needs though to have uniform Schauder estimates on these domains, that is, an analogue of the basic estimate of <cit.>.  As in <cit.>, we can still use a barrier function to get good control outside some large compact set, which helps in determining weights for which we will have an isomorphism theorem.  In particular, when there are maximal rank cusps, we can only get isomorphism theorems for unbounded weight functions in the cusp ends of finite volume, making them unsuitable for a possible application of the inverse function theorem.  This is the reason why maximal rank cusps are excluded in Theorem <ref>, as well as cusps of rank 2 when n=4.   


Unfortunately however, the part of the argument of <cit.> giving uniform control on domains is very particular to the model hyperbolic space and does not apply to our setting.   On the other hand, the approach of Lee <cit.>, which passes through a careful study of Fredholm properties of geometric operators (see also the work of Mazzeo and Melrose <cit.> on that matter), could possibly be adapted to our setting, but would definitely require a substantial amount of work.   We found instead a more direct route to obtain the isomorphism theorem.  Our strategy is to suppose that the Schauder estimates degenerate as the domains exhaust the manifold to derive a contradiction by obtaining a sequence of tensor fields that converges to a nonzero element of the (trivial) kernel of the linearized operator.  One delicate aspect in this argument is that because of the cusps of intermediate rank,  while the corresponding hyperbolic metric has constant curvature, it is not of bounded geometry since its injectivity radius vanishes.  As in <cit.>, near a cusp, one can nevertheless get uniform local Schauder estimates by passing to a suitable cover, that is, quasi-coordinates in the terminology of  <cit.>.  On the other hand, on say a convex co-compact hyperbolic manifold, our isomorphism theorem gives a shorter and simpler proof of the perturbation result of <cit.>.  The price to pay with this method however is that one needs the  corresponding weighted Hölder spaces to lie in L^2.  This restricts the regularity of the boundary metric encoded as k in the statement of the theorem.  


Even if the more detailed statement of our main result, Theorem <ref> below, gives some control on the behaviour of the metric at infinity, especially away from the cusps, one could ask if stronger regularity results hold.  For instance, by <cit.>, we know that Poincar-Einstein metrics admit a polyhomogeneous expansion at infinity, so one could ask more generally if the geometrically finite Poincar-Einstein metrics of Theorem <ref> admit a polyhomogeneous expansion with respect to the compactification of Mazzeo-Phillips.  In the hope of making progress on this question and inspired by <cit.>, we are planning in a subsequent work to develop a pseudodifferential calculus adapted to geometrically finite Poincar-Einstein metrics.  Additionally, we expect it should be possible to relax the hypothesis of compactly supported perturbations in our result to perturbations lying in a Hölder space adapted to the foliated cusp metric at infinity.  This raises the possibility of understanding the extent to which the results of conformally compact geometry can be adapted to this more general conformally foliated cusp setting.


The remainder of this paper is structured as follows.  In <ref> we introduce the class of (0-)-metrics which are conformally related to geometrically finite hyperbolic metrics.  In <ref> we introduce geometrically finite hyperbolic metrics and their compactification to a suitable manifold with corners.  Suitable function spaces for our analysis are defined in <ref>, and <ref> states and proves a uniform boundary Schauder estimate for a certain exhaustion of M.  In <ref> we prove an isomorphism theorem for Laplace operators, and study the application to our nonlinear problem.  We then prove Theorem <ref> in <ref>.


Acknowledgements.  We are happy to acknowledge useful conversations with Robin Graham, Jack Lee and Rafe Mazzeo.  EB gratefully acknowledges the hospitality of the CIRGET laboratory and the support of the Centre de Recherches Mathématiques through their CRM Simons Professor program, and the support of a Simons Foundation grant (#426628, E. Bahuaud).  FR was supported by NSERC and a Canada Research chair.



§ (0-)-METRICS
 

In this section we introduce a Lie algebra of vector fields on a suitable compact manifold with corners and associated metrics that are conformal to geometrically finite hyperbolic metrics.


Let X be a compact manifold with corners of dimension n.  Let n_ir and n_mr be non-negative integers and set n_c = n_ir + n_mr.  Suppose that H_0,H_1,…, H_n_c is an exhaustive list of the boundary hypersurfaces of X such that 

    H_i∩ H_j=∅ ⟺  i j,   ∀  i,j∈{1,…,n_c}.

Moreover, the first n_ir boundary hypersurfaces distinct from H_0 satisfy

    H_0∩ H_j∅  ∀  j ∈{1, …, n_ir},

while the final n_mr boundary hypersurfaces satisfy

    H_0 ∩ H_j = ∅  ∀  j ∈{n_ir+1, …, n_c}.

Hereafter we use the convention that a greek index takes values within the index set {1, …, n_ir}.  Thus, X only has codimension 2 corners and the boundary hypersurfaces meeting at these corners are themselves compact manifolds with boundary with 

    H_0= ⋃_1 ≤α≤ n_ir H_α∩ H_0,       H_α= H_α∩ H_0,  ∀α∈{1, …, n_ir}.

Let x_0, x_1,…, x_n_c be boundary defining functions for H_0,…, H_n_c respectively.  Without loss of generality, assume that for each i, x_i≡ 1 outside a tubular neighbourhood of H_i so that for i≥ 1, x_i=1 near H_j for j≥ 1 with j i.  


For α∈{1, …, n_ir}, suppose that there is a smooth foliation _α of H_α compatible with the boundary defining function x_0 in the sense that the level sets of x_0|_H_α near H_0∩ H_α are tangent to the foliation, that is, any leaf intersecting them is in fact contained in them.  This implies in particular that _α restricts to give a foliation _0α:= _α|_ H_α   on H_α.  Let us denote by  the collection of foliations (_1,…, _n_ir) on the boundary hypersurfaces H_1,…, H_n_ir of X.  Now, recall that on X, the Lie algebra of b-vector fields _b(X) consists of smooth vector fields ξ∈(X;TX) which are tangent to all boundary hypersurfaces of X.  




The Lie algebra _0-(X) of (0-)-vector fields on X consists of vector fields ξ∈_b(X) such that 

    ξ|_H_0=0; 
    ξ|_H_α∈(H_α;T_α)   α∈{1, …, n_ir}; 
    ξ x_i∈ x_i^2(X)      i ∈{1, …, n_c}.




Since these conditions are clearly preserved by the Lie bracket, we see that _0-(X) is indeed a Lie subalgebra of _b(X) and (X;TX).  As the notation suggests, the definition of _0-(X) depends on the collection of foliations .  More subtly, it depends on the choice of the boundary defining functions x_1,…, x_n_c through the condition (<ref>). 


Near H_0, but away from the other boundary hypersurfaces, a (0-)-vector field behaves like a 0-vector field of  <cit.>.  Near H_α, but away from H_0, it looks instead like a foliated cusp vector field (or _i-vector field) of <cit.>, the foliated version of fibered cusp vector fields of <cit.>.   Near H_i, i ≥n_ir +1, it looks like a cusp vector field (a fibred cusp vector field associated to a trivial fibration over a point).  At the corner H_0∩ H_α, the (0-)-vector fields are somewhat of a hybrid version of the 0-vector fields of <cit.> and the foliated cusp vector fields of <cit.>.  This can be described more precisely in local coordinates.  Given p∈ H_0∩ H_α, choose a coordinate chart φ: →Ω⊂^n sending p to 0 with 

    φ= (x_0,x_α,y_1,…, y_b_α,z_1,…, z_f_α)

where x_0 and x_α are the boundary defining functions chosen above and the coordinates (y,z) are such that that the distribution T_α associated to the foliation _α at x_α=0 is spanned by
/ z_1,…,/ z_f_α.  In such a coordinate chart, (0-)-vector fields are of  the form

    a x_0x_α/ x_0 + bx_0x_α^2/ x_α+ ∑_j=1^b_α c_jx_0x_α/ y_j + ∑_k=1^f_α d_k x_0/ z_k

for a,b, c_1,…,c_b_α, d_1,…, d_f_α  arbitrary smooth functions.  Indeed, the factor x_0 in each term ensures that condition (<ref>) is satisfied, while the factors of x_α and x_α^2 ensure that conditions (<ref>) and (<ref>) are satisfied.  


This shows that _0-(X) is a locally free (X)-module of rank n.  Let  ^0-TX→ X be the vector bundle with fiber above p∈ X given by 

    ^0-T_pX:= _0-(X)/ _p_0-(X)
  
where _p⊂(X) is the ideal of smooth functions vanishing at p.  By the Serre-Swan theorem, this vector bundle comes with a canonical morphism of vector bundles 

    𝔞: ^0-TX→ TX
 
inducing the identification

    (X;^0-TX)=_0-(X).

The map (<ref>) induces an isomorphism ^0-TX|_X∖ X≅ TX|_X∖ X, but fails to be an isomorphism on X.  We remark that the vector bundle ^0-TX→ X is naturally a Lie algebroid with anchor map given by (<ref>) and Lie bracket on its sections induced by the identification (<ref>), although this fact will not be used in this paper.


In the terminology of <cit.>, _0-(X) is a Lie structure at infinity.  There is therefore a natural class of metrics associated to it.  We say a (0-)-metric on X∖ X is a Riemannian metric corresponding to the restriction of a bundle metric on ^0-TX to T(X∖ X) via the identification
T(X∖ X)≅^0-TX|_X∖ X induced by the anchor map (<ref>).  


In the coordinate chart (<ref>), an example of (0-)-metric is given by

    dx_0^2/(x_0x_α)^2+ dx_α^2/(x_0x_α^2)^2+ ∑_j=1^b_αdy_j^2/(x_0x_α)^2+ ∑_k=1^f_αdz_k^2/x_0^2.



Since (0-)-metrics come from a Lie structure at infinity, we know from <cit.> that they are complete of infinite volume and that they are all quasi-isometric to each other.  Furthermore, it is shown in <cit.> that their curvature and all its derivatives are bounded.  On the other hand, one can check directly in local coordinates that the injectivity radius is positive, so that (0-)-metrics are of bounded geometry.  


The class of metrics we are interested is not quite the one of (0-)-metrics, but a conformal cousin.  Define a (0-)_c-metric g_(0-)_c on X∖ X to be a metric of the form

    g_(0-)_c:= (∏_i=1^n_cx_i)^2 g_0-

for some (0-)-metric g_0-.  In the coordinate chart (<ref>), an example of (0-)_c-metric is given by

    dx_0^2/x_0^2+ dx_α^2/(x_0x_α)^2+ ∑_j=1^b_αdy_j^2/(x_0)^2+ ∑_k=1^f_αx_α^2dz_k^2/x_0^2.


Despite this extra conformal factor, a (0-)_c-metric is still complete of infinite volume, but it becomes of finite volume outside a tubular neighbourhood of H_0.  More importantly, the injectivity radius is no longer positive and the curvature is typically not bounded.  Near H_0 but away from H_i for i≥ 1, it still looks like a 0-metric.  Near H_α for α∈{1, …, n_ir} but away from H_0, it looks instead like a foliated cusp metric of <cit.>, and near H_i for i ≥ n_ir+1 it looks like a cusp metric.


In the next section we explain how geometrically finite hyperbolic metrics provide examples of (0-)_c-metrics.



§ GEOMETRICALLY FINITE HYPERBOLIC METRICS



In this section we define the class of hyperbolic metrics of interest and fix notation.  Our description of a suitable compactification of these hyperbolic metrics to a manifold with corners was first stated by Mazzeo and Phillips <cit.> and we refer the reader to a very thorough discussion there.  Compare also <cit.> and <cit.>.


We consider the simply connected hyperbolic space of constant curvature -1, ^n, represented by the Poincaré ball model.  In what follows, Γ will be a discrete, torsion-free group of isometries acting properly discontinuously on ^n.  Recall that (nontrivial) isometries of ^n are classified as elliptic, parabolic or hyperbolic if they fix 0, 1 or 2 points on the boundary sphere at infinity, ^n-1_∞.  Our interest will be when Γ is geometrically finite with cusps, which as we explain will entail that Γ contains parabolic elements and that an appropriate quotient of a subset of ^n may be written as a union of a compact region and finitely many disjoint cusp neighbourhood ends.  When n > 3, this is more general that requiring that Γ possesses a fundamental domain for the action bounded by totally geodesic hyperplanes and portions of the sphere at infinity, see <cit.>.


Given any p ∈^n, the set of accumulation points Λ_Γ (in ^n = ^n ∪^n-1_∞) of the orbit Γ· p is independent of the choice of p and is called the limit set of Γ.  The action of Γ on ^n extends to a properly discontinuous action on Ω_Γ = ^n-1_∞∖Λ_Γ, and Γ acts properly discontinuously when we adjoin this conformal boundary M := (^n∪Ω_Γ) / Γ.  Note that  is now a manifold with noncompact boundary.  It is this latter manifold we now decompose further.


Now fix a parabolic element γ∈Γ.  Let p ∈^n-1_∞ be the fixed point of γ.  This point gives rise to a cusp.  Let Γ_p be the subgroup of elements of Γ that fix p.   Consider the upper-half space model of hyperbolic space ^n={(x,y^1,…,y^n-1: x > 0 } with the fixed point p placed at infinity x=+∞.  Each level set of x, E_a := { x = a } is a horosphere on which the elements of Γ_p acts by euclidean isometries, and as such there is a maximal normal free Abelian subgroup of Γ_p with finite index f; the number f is called the rank of the cusp.  We distinguish two cases: when the cusp has maximal rank n-1 and when the cusp has intermediate rank 1 ≤ f < n-1.  


As explained in <cit.>, it is possible to find a maximal subspace ^f⊂^n-1 = E_1, 1 ≤ f ≤ n-1 that is mapped to itself under Γ_p.  For a certain polyhedral fundamental domain K ⊂^f, the fundamental domain for the action of Γ_p on ^n-1 becomes ^n-1-f× K, where the action on the ^n-1-f factor is through rotation.  The quotient ^n-1 / Γ_p is then the total space of a flat vector bundle over a compact manifold K/Γ_p.  Furthermore, ^+  ×^n-1-f× K is a fundamental domain for the action of Γ_p on ^n.  Returning to the splitting ^n = ^+  ×^n-1-f×^f, define

    C_p(R) := { (x,y,z) ∈ [0,∞) ×^n-1-f×^f: x^2 + |y|^2 ≥ R}⊂^n.

The set C_p(R) is invariant under Γ_p and convex with respect to the hyperbolic metric.  In a similar way we define C_p(R) for all other parabolic fixed points.  For each parabolic fixed point p, there exists R > 0 sufficiently large so that


  * C_p(R) ⊂^n ∪Ω_Γ and

  * For any γ∈Γ∖Γ_p, γ C_p(R) ∩ C_p(R) = ∅.


As a consequence, C_p(R) descends to a set 𝒞_p = (∪_γγ C_p(R)) / Γ which has interior isometric to C_p(R) / Γ_p.  We call this the standard cusp region 𝒞_p associated to the orbit of p; <cit.>.  Thus we say M := ^n / Γ is geometrically finite if = (^n ∪Ω_Γ) / Γ has a decomposition into the union of a compact set 𝒞_0 and a finite number n_c of standard cusp regions, with n_c ≥ 1:

    = (^n ∪Ω_Γ) / Γ = 𝒞_0 ∪( ⋃_j=1^n_c𝒞_p_j).

Note that the compact set 𝒞_0 contains points on ∂=Ω_Γ/Γ.  Moving forward we assume that the enumeration of the cusps is now fixed, and with n_ir intermediate rank cusps followed by  n_mr cusps of maximal rank so that n_c = n_ir + n_mr.
We now describe the compactification by Mazzeo-Phillips <cit.> of  to a manifold with corners.


We start with the compactification of a single intermediate rank cusp region, 𝒞_p by working with a representative C_p.  Introduce coordinates (x,y^1,…,y^n-1-f,z^1,…,z^f) ∈^+ ×^n-1-f× K, and recall that the cusp is at x=+∞.  In each fibre z = z_0, invert in the unit sphere through the map (u,v^i) = (x/x^2 + |y|^2, y^i/x^2 + |y|^2).  This places the cusp at (u,v) = (0,0) and the cusp region of the manifold is now diffeomorphic to ^n-f_+× K.  In this region the hyperbolic metric is expressed as 

    h = du^2 + ∑_i=1^n-1-f (dv^i)^2 + (u^2+|v|^2)^2 ∑_i=1^f(dz^i)^2/u^2.



We blow up {0}×{0}×^f in ^+×^n-1-f×^f  by introducing polar coordinates r = √(u^2 + |v|^2), ϕ∈^n-1-f_+, with ϕ^0 = u/r > 0, ϕ^i = v^i/r.  It will be convenient to write the coordinate ϕ^0 as ρ instead.  Thus the blown up cusp neighbourhood is diffeomorphic to ^+_r ×^n-1-f_+×^f and we may compactify to a manifold with corners by adjoining the boundary faces r = 0 (the cusp face) and ρ = 0 (the 0-face), obtaining [0,∞)×^n-1-f_+×^f, where ^n-1-f_+ is the closure of _+^n-1-f in ^n-f.  Notice that Γ_p acts naturally on this space.
Passing to the quotient, we have compactified 𝒞_p by

    𝒞_p := [0,)_r × ((^n-1-f_+)_ϕ×^f)/Γ_p,

a hemisphere bundle over  [0,)_r×^f/ Γ_p.  
 We use {w^1, …, w^f} to represent local coordinates on ^f/ Γ_p.  In a local trivialization of the hemisphere bundle on ^f/Γ_p,  hyperbolic metric can then be written

    h|_𝒞_p = dr^2/r^2 ρ^2 + dϕ^2/ρ^2 + r^2/ρ^2 dw^2


where dw^2 is a flat metric and d ϕ^2 is the round metric on the (n-1-f)-sphere.  Observe that the function u = rρ descends to [0,)_r × ((^n-1-f_+)_ϕ×^f)/Γ_p and provides a total boundary defining function at this end.


If the cusp is of maximal rank, then the fundamental domain for Γ_p intersects the level set E_1 in a compact polyhedron K ⊂^n-1 that is a fundamental domain for the action of Γ_p on ^n-1.  We obtain a compact flat manifold N = ^n-1 / Γ_p, and then 𝒞_p is diffeomorphic to [R,∞)× N.  Set r = 1/x in this neighbourhood and then adjoin the face r = 0 to obtain the compactification

    𝒞_p=[0,1/R]_r× N.
 
In the region 𝒞_p, the hyperbolic metric may be written

    h|_𝒞_p = dr^2/r^2 + r^2 g_N

with g_N a flat metric on N.


Finally, near any point in 𝒞_0, the compactification is already provided by M=(^n∪Ω_Γ)/Γ with boundary face 
given by Ω_Γ/Γ.  Away from the cusps, say for U an open set in Ω_Γ/Γ with closure not intersecting the cusps, the metric near this boundary can be put in the form

    h= dρ^2+ h_U(ρ)/ρ^2      [0,δ)_ρ× U

with h_U(ρ) a smooth family of metrics on U parametrized by ρ∈ [0,δ) with ρ=0 corresponding to the conformal boundary at infinity.  Note that ρ is a special boundary defining function here since |dρ|^2_ρ^2h≡ 1 near ρ=0.   Notice however that in the cusp neighbourhood 𝒞_p, the manifold Ω_Γ/Γ is itself compactified by adding a boundary, namely 
(^n-1-f_+×^f)/Γ_p.  




Collecting these observations, we have thus obtained a compactification of = (^n ∪Ω_Γ) / Γ to a smooth manifold with corners  with boundary hypersurfaces H_0, H_1, …, H_n_c, where 
H_0 = Ω_Γ/Γ is the closure of Ω_Γ/Γ in , and H_i={0}× (^n-f-1×^f)/Γ_p is the i-th cusp face defined by { r_i = 0} in 𝒞_p_i for some p_i∈^n-1_∞.  Note that for each intermediate rank cusp face H_α, H_0 ∩ H_α is codimension two corner, and that H_i ∩ H_j = ∅ for i ≠ j.  Each compactified intermediate rank cusp neighbourhood 𝒞_p_α is the total space of a flat bundle,

    Φ_i: 𝒞_p_α⟶ D_α

over a compact flat manifold D_α that yields a cusp of rank f_α =  D_α.  Since the bundle (<ref>) is flat, its horizontal distribution is integrable, so induces a foliation on H_α that we will denote _α.      In terms of the decomposition (<ref>), a leaf of _α is the image under the quotient map by the action of Γ_p_α of a submanifold of the form

    {0}×{q}×^f_α⊂{0}× (^n-1-f_α_+)_ϕ×^f_α

for some q∈ (^n-1-f_α_+)_ϕ.


In what follows we may assume that the r_i's are globally defined by smoothly truncating its value to be 1 outside of its cusp neighbourhood.  We continue to let ρ be a defining function for H_0 which corresponds to ρ  in the coordinate chart of (<ref>) for each α.  The product 

    σ = ρ∏_i=1^n_c r_i
 
furnishes a smooth global boundary defining function, and for sufficiently small > 0 the superlevel sets

    M_ = { p: σ(p) ≥}

furnish a compact exhaustion of ∖ by smooth compact manifolds with boundary.


We now connect these hyperbolic metrics to the metrics defined in the previous section.

The hyperbolic metric on ∖ is a (0-)_c-metric with respect to the (0-)-Lie structure at infinity on 
 induced by boundary defining functions ρ,r_1,…, r_n_c and the collection of foliations =(_1,…, _n_ir).


It suffices to notice that, near each H_α for α∈{1, n_1, …, n_ir}, the hyperbolic metric is precisely of the form (<ref>) by (<ref>), with similar observations for the other hypersurfaces.


Finally, we remark that as a (0-)_c-metric, the hyperbolic metric is very special, since it has constant sectional curvature, which implies that its curvature and all derivatives are bounded.  However it is not of bounded geometry since its injectivity radius is zero.  Thus when obtaining estimates in weighted Hölder spaces we will pass to a cover of positive injectivity radius and consequently of bounded geometry, see <ref>.       


In the next section we extend the notion of conformally compact asymptotically hyperbolic metric to a broader class modeled on these metrics.



§ FUNCTION SPACES
 

Let (M,h) be a geometrically finite hyperbolic metric with cusps as described in the previous section.  Recall that an enumeration of the cusps is fixed.  Through the paper we write Σ^2(M) for the bundle of symmetric 2-tensors and Σ^2_0(M) for the elements of Σ^2(M) that are also trace-free with respect to the metric.  


For the smooth tensor bundle of (p,q)-tensors, E := (TM)^⊗ p⊗ (T^*M)^⊗ q⟶ M, we have the usual space of L^2 sections, defined as the completion of compactly supported smooth sections  with respect to the L^2 norm

    u_L^2 := ( ∫_M |u|_h^2 dv_h )^1/2.



To describe weighted versions of these spaces, encode weighting with respect to a vector μ=(μ_0,μ_1,…, μ_n_c) ∈^1+n_c by 

    σ^μ := ρ^μ_0 r_1^μ_1… r_n_c^μ_n_c.

Note that we write inequalities like μ≥ 0 as shorthand for μ_i ≥ 0 for all i = 0, …, n_c.  Define

    L^2_μ(M;E) = σ^μ L^2(M;E)
 
with norm u _L^2_μ := σ^-μ u _L^2.


The following criterion will be useful to determine  when a section of E is square integrable.

 
Suppose a continuous section u: M → E satisfies |u|_h ≤ C σ^μ for some C > 0.  If 

    μ_0 > n-1/2,  μ_j > -f_j/2,  ∀ j ∈{1, …, n_c} ,

where f_j is the rank of the j-th cusp, then u ∈ L^2(M;E).


It suffices to check that  the L^2 norm is finite both in the neighbourhood _0, where σ = ρ and in any cusp neighbourhood _p_j where σ = ρ r_j.  Here we outline the calculation for the cusp neighbourhood as the calculation in the regular neighbourhood is simpler.  In view of the expression for the metric given in equation (<ref>), one may compute an expression for the volume form and then

    ∫__p_j |u|_h^2 dv_h    ≤ C ∫_0^* ∫_0^* ρ^2 μ_0 r_j^2 μ_jr_j^f_j-1/ρ^n  dr_j dρ≤ C ∫_0^*  ρ^2 μ_0 - n dρ∫_0^* r_j^2 μ_j+f_j-1 dr_j.

This integral is finite if μ_0 > (n-1)/2 and μ_j > -f_j/2.



For any non-negative integer k, let C^k(M,E) be the space of  k-times continuously differentiable sections u of E such that the norm

    u_C^k(M,E) := ∑_ℓ=0^k sup_M | (∇^h)^ℓ u |_h

is finite. 


Before introducing the Hölder quotient, by a path we will mean any piecewise C^1 map γ: [0,1] → M.  The length of a path is given by

    len(γ) = ∫_0^1 | γ'(s) |_h   ds.

In what follows we let P_γ: E|_γ(0)⟶ E|_γ(1) denote  parallel transport of sections of E along γ with respect to the metric h.


For any non-negative integer k and α∈ (0,1] we define the (k,α)-norm on C^k sections of E by

    u _k,α =  u _C^k,α(M;E) := [u]_k,α +  u _C^k(M;E) ,

where the Hölder seminorm is defined by

    [u]_k,α := sup{|P_γ(∇^k u( γ(0) ) ) - ∇^k u( γ(1) ) |_h /len( γ )^α: γ γ(0) ≠γ(1) }.

The space C^k,α(M;E) is then defined as the subspace of C^k sections of E for which the norm ·_k,α is finite.  


Note that 

    ∇^h: C^k,α(M;E)   C^k-1,α(M;T^*M ⊗ E)

is a bounded operator.  In view of the metric decomposition of equation (<ref>) in each cusp neighbourhood described in the previous section, requiring that a section lie in a Hölder space imposes some restriction on the asymptotic behaviour of that section at each boundary hypersurface.  For example, for a function u: M ⟶ restricted to the j-th cusp neighbourhood, boundedness of sup |∇^h u|_h is equivalent to separate boundedness of the (0-)_c-derivatives

    rρ∂ u/∂ r, ρ∂ u/∂ϕ, ρ/r∂ u/∂ w,

where we abbreviated r = r_j.  If ρ/r∂ u/∂ w is bounded, then ρ∂_w u must vanish as r → 0 and ρ > 0, and thus ∂_w u extends to be 0 along r=0 when ρ > 0.  Thus, if u has a C^1 extension to , then the restriction of u to a leaf of _j gives a constant function.


We additionally define a weighted Hölder space C^k,α_μ(M;E) := σ^μ C^k,α(M;E) with norm

    u_k,α,μ := σ^-μ u _k,α.
 


The following lemma gives some of the basic properties of these Hölder spaces.


Let k ∈_0, α∈ [0,1].  Let E_1, E_2 denote tensor bundles over M.


  * The hyperbolic metric h, its inverse h^-1 and any covariant derivative of curvature ∇^ℓR^h lie in ⋂_m ∈ C^m,α_0.

  * Contracting with h or h^-1 preserves weight and regularity.

  * Pointwise tensor product induces a continuous map

    C^k,α_μ(M;E_1) × C^k,α_μ'(M,E_2)  C^k,α_μ + μ'(M; E_1 ⊗ E_2).


  * The covariant derivative induces a continuous map

    ∇^h: C^k,α_μ(M;E_1)   C^k-1,α_μ(M;T^*M ⊗ E_1).


  * For any non-negative weight μ and any ℓ∈_0, sup_M | (∇^h)^ℓσ^μ | is bounded.  In particular, for two weights μ_1 and μ_2 with μ_1≥μ_2, there is a continuous inclusion

    C^k,α_μ_1(M;E_1) ⊂ C^k,α_μ_2(M;E_1).




Items (a)–(d) are straightforward.  


Item (e) stems from our particular choice of total boundary defining function given in the previous section.  Recall that σ and the choice of coordinates in each cusp neighbourhood is adapted to the geometry.  In particular in the j-th cusp neighbourhood, σ = ρ r_j is constant along the leaves of _j, thus the action of the singular vector field ρ/r∂/∂ w on σ always vanishes, which need not be true for an arbitrary total defining function.  Thus for any ℓ∈_0 and for any non-negative weight, sup_M | (∇^h)^ℓσ^μ | is bounded.  For weights μ_1 and μ_2 such that μ_1≥μ_2, this implies that there is a continuous inclusion 

    C^k,α_μ_1(M;E_1) ⊂ C^k,α_μ_2(M;E_1).




With these preliminaries in hand, we now extend the notion of conformally compact asymptotically hyperbolic metric.  For motivation we briefly review two well-known types of metrics.  


First, if (,) is a smooth compact manifold with boundary, and ρ is a smooth boundary defining function, then the conformally compact metric g = ρ^-2 is always asymptotically negatively curved near ∂ M. If the conformally invariant condition that |dρ|^2_ρ^2 g = 1 on ∂ M is satisfied, then all sectional curvatures of g tend to -1 plus corrections of order ρ.  Moreover (M,g) is complete and of bounded geometry.  The facts above are due to Mazzeo <cit.>.


Now given a hyperbolic metric h  as in equation (<ref>) with accompanying open subset U ⊂Ω_Γ / Γ away from the cusps, the discussion above shows that we may take perturbations by symmetric 2-tensors “of the same order” as the metric with respect to the defining function, subject only to |dρ|^2_ρ^2 g = 1 at ρ = 0.  In particular, any “tangential” perturbation of the form

    g = dρ^2+ h_U(ρ)/ρ^2+ e_U(ρ)/ρ^2

will be asymptotically hyperbolic, where e_U(ρ) is a smooth family of small 2-tensors on U parametrized by ρ.  Note that |e_U|_g = O(ρ^2).


Second, consider a hyperbolic cusp metric in a compactified neighbourhood, as in equation (<ref>).  This metric has bounded curvature only when the induced metric on the leaves of _i is flat.  Thus in these neighbourhoods, we must always take perturbations which vanish to some positive order.


Now suppose that (M,h) := (^n / Γ,h) is a geometrically finite hyperbolic metric with n_ir intermediate rank cusps and n_mr maximal rank cusps, and consider the compactification  described in the previous section.   We say that a Riemannian metric g on ∖ is asymptotically hyperbolic with cusps modeled on (M,h) if there is a weight vector μ∈^1+n_c with μ_0 ≥ 0 and μ_i > 0 for i = 1, …, n_c so that g is a (0-)_c-metric of the form

    g = h + e,

where e symmetric (0-)_c 2-tensor with e ∈ C^k,α_μ(M;Σ^2(M)), and |dρ|^2_ρ^2 g = 1 on H_0.  When necessary we will write that g is (k,α,μ)-asymptotically hyperbolic to emphasize the parameters involved, always keeping in mind the fixed geometrically finite hyperbolic reference metric.


As the leading part of g is h, it will be useful to perform analysis of g in the function spaces defined with respect to h.  Note that g is quasi-isometric to h.  We additionally have

 Let g be a (k,α,μ)-asymptotically hyperbolic metric with cusps as above.  Then for ℓ≤ k,

    ∇^g: C^ℓ,α_μ(M;E)  C^ℓ-1,α_μ(M;T^*M ⊗ E)

is bounded.


First, in any system of local coordinates, write the Christoffel symbols of h and g as Γ̂_ij^p and Γ_ij^p respectively.  Recall that 

    A_ij^p = Γ̂_ij^p - Γ_ij^p

define the components of a well-defined (1,2)-tensor field.  Evaluating this tensor in h-normal coordinates at a point allows us to see that

    A_ij^p = -1/2 g^pm ( ∇^h_i e_jm + ∇^h_j e_im - ∇^h_m e_ij).

In particular if g is (k,α,μ)-asymptotically hyperbolic then ∇^h e is  C^k-1,α_μ, and thus A is  C^k-1,α_μ.


If u is a (p,q)-tensor, then the formula for the covariant derivative with respect to g may be expressed in terms of the covariant derivative of h and various contractions with A.  We write this abstractly as

    ∇^g u = ∇^h u + A * u,

where A*u denotes linear combinations of contractions of A with u. Thus if u ∈ C^ℓ,α_μ(M;E), each term on the right hand side of this equation lies in C^ℓ-1,α_μ(M;E), since min{ℓ-1,k-1} = ℓ-1 when ℓ≤ k.


As mentioned above, a generic cusp metric has unbounded curvature when the link is not flat.  As we now prove, since our asymptotically hyperbolic metrics are hyperbolic at leading order and we always take the cusp weight such that μ_i > 0, i>0, we obtain metrics with bounded curvature.

  Let k ≥ 2 and define 𝒰 := { e ∈  C^k,α_μ(M;Σ^2(M)): h+e  }.  Then for e ∈𝒰, 
 g = h + e has bounded curvature and the nonlinear operator

    𝒰    C^k-2,α_μ(M; Σ^2(M)) 
    
     e    quebec(h+e) + (n-1)(h+e)

is a smooth map of Banach manifolds.
  

Consider the Riemann curvature (1,3)-tensor of g, ^g, which we may write  abstractly

    ^g = ∂Γ + Γ * Γ,

where Γ represents the Christoffel symbols with respect to g.  Interpolating the Christoffel symbols Γ̂ of the reference hyperbolic metric h and using A as the difference tensor as in the proof of the previous proposition, one obtains abstractly

    ^g    = ^h+ ∂ A + A*Γ̂ + A*A 
       = ^h+∇^h* A + A*A,
 
where in the second equality we express the tensor in h-normal coordinates at a point, in order to recognize a h-covariant derivative, and the asterisk (*) indicates linear combinations of the quantities indicated, whose explicit form is unimportant for what follows.


As in the previous proof, A ∈ C^k-1,α_μ, and thus ∇^h * A ∈ 
C^k-2,α_μ, and A*A ∈ C^k-1,α_2μ.  This shows that g has bounded curvature.


Contracting equation (<ref>) and adding a factor of (n-1) g to both sides we obtain

    (g) + (n-1)g    = (h) + (n-1) (h+e) + ∇^h * A + A*A 
       = (n-1) e+∇^h * A + A*A,

which lies in C^k-2,α_μ once more.  The smoothness of  follows from the fact that the Ricci tensor is a polynomial contraction of the inverse metric and its first two derivatives.


Using this proposition and similar estimations, one may check a similar mapping property for the gauge-adjusted Einstein equation defined in equation (<ref>).
  
The nonlinear operator

    𝒰×𝒰    C^k-2,α_μ(M; Σ^2(M)) 
    
     (e_1,e_2)    quebecQ(h+e_1,h+e_2)

is a smooth map of Banach manifolds.




§ BOUNDARY SCHAUDER ESTIMATES FOR THE EXHAUSTION



The proof of the isomorphism theorem for operators of the form Δ^h + K (with K a constant) requires boundary Schauder estimates applied over an exhaustion of M by the superlevel sets

    M_ := { p ∈ M: σ(p) ≥}

defined by the total boundary defining function σ.  This section is devoted to the proof of the following uniform Schauder estimate:
 
Let ω∈ C^k,α(M_;E) be a solution to (Δ + K) ω = τ in M_, ω|_∂ M_ = 0.  Then for  sufficiently small,

    ω_k,α; M_≤ C ( τ_k-2,α;M_ + ω_0,0;M_ ),

where C = C(n,α,k,K) is independent of .


In order to establish this result, we follow the strategy used by Cheng-Yau <cit.> and Graham-Lee <cit.>.  It suffices to estimate the Hölder norm of u locally in small balls of a uniform size both in the interior and on the boundary, where the metric and hence the coefficients of the operator Δ + K, are appropriately controlled.  In a large compact set away from all of the boundary hypersurfaces, the estimate follows from classical elliptic theory.  We thus explain how to estimate in a cusp neighbourhood, where the injectivity radius shrinks to zero, and near H_0.


Fix a cusp neighbourhood 𝒞_p where f is the rank of the cusp, as in <ref>.  The fact that the injectivity radius is zero is due to the side identifications of the polygon K using Γ.  Thus instead of working directly in the cusp neighbourhood, we will “unroll” the cusp and work on the universal cover.  Recall that M_ is locally defined near the cusp relative to the coordinates (u, v^i, z^j) introduced before equation (<ref>) by the inequality u ≥ (in these coordinates the cusp is located at u = 0, v = 0).  Pulling back the tensor ω:M → E to this region amounts to considering a periodic extension in the z^j variables.  As the definition of Hölder spaces essentially measure local quantities and our definition of Hölder spaces are defined geometrically, the Hölder norm of the pullback to the universal cover coincides with the Hölder norm on M.


Fix a point p_0 ∈∂ M_∩ C_p.   Locally we may write p_0 = (, v^i_0, z^j_0).  We first consider the case where |v^i_0| ≤ C.    Consider ^n = _s ×^n-1-f_p ×^f_q and the euclidean reference half-ball B_+ := {(s,p^i,q^j): s^2 + | p |^2 + |q|^2 < 1,    s ≥ 0 }, with B' denoting the half-ball with radius 1/2.  Define a map

    Ψ_p_0: B_+ →Ψ_p_0(B_+)

by

    (u,v^i,z^j) = Ψ_p_0(s, p^i, q^j) = (  e^s, v_0^i +  p^i, z_0^j + 1/ q^j ).
 
The hyperbolic metric of (<ref>) pulls back via Ψ_p_0 to

    Ψ^*_p_0 h = ds^2 + e^-2s∑ (dp^i)^2 + e^-2s( e^2s +  |1/ v_0 + p|^2 )^2 ∑ (dq^j)^2.

At least for |v^i_0| ≤ C, the coefficients of any number of derivatives of this metric are uniformly bounded on B_+.  Moreover, there is a uniform positive lower bound on the eigenvalues of the inverse of Ψ^*_p_0 h.  Thus the Laplacians associated to this metric are uniformly elliptic with uniformly (in ) Hölder continuous coefficients.


Now pulling back (Δ + K) ω = τ to B_+, we may apply a standard local boundary Schauder estimate <cit.> to obtain an estimate of the form

    Ψ^*_p_0ω_k,α;B_+ ∩ B'≤ C ( Ψ^*_p_0τ_k-2,α;B_+ + Ψ^*_p_0ω_0,0;B_+),

where C is independent of  and p_0.  Note that the Hölder spaces referenced here are with respect to the background euclidean metric in (s,p,q)-coordinates. However it is straightforward to check that this implies that an estimate with Hölder spaces with respect to Ψ^* h holds with a different constant since the Ψ^* h and the euclidean metric and all derivatives are uniformly comparable in this chart.  We conclude that

    ω_k,α; Ψ_p_0(B') ≤ C ( τ_k-2,α;M_ + ω_0,0;M_ ),

which yields the required local estimate when p_0 is close to the cusp (as quantified by the inequality |v^i_0| ≤ C).


We now consider the case where p_0 = (, v^i_0, z^j_0) ∈∂ M_∩ C_p and  < |v^i_0| < 1.  A slightly different rescaling is required.  Consider ^n = _s ×^n-1-f_p ×^f_q and the euclidean reference half-ball B_+ := {(s,p^i,q^j): s^2 + | p |^2 + |q|^2 < 1,    s ≥ 0 }, with B' denoting the half-ball with radius 1/2.  Define the map

    Ψ_p_0: B_+ Ψ_p_0(B_+)

by

    (u,v^i,z^j) = Ψ_p_0(s, p^i, q^j) = (  e^s, v_0^i +  p^i, z_0^j + /^2 + |v_0|^2 q^j ).
 
The hyperbolic metric of (<ref>) pulls back via Ψ_p_0 to

    Ψ^*_p h = ds^2 + e^-2s∑ (dp^i)^2 + e^-2s( e^2s + |v_0|^2/^2 + 2 v_0/· p + |p|^2 )^2/(1 + |v_0|^2/^2)^2 ∑ (dq^j)^2.

Once more, the coefficients of any number of derivatives of this metric are uniformly bounded on B_+, independently of , and inverse metric has eigenvalues uniformly bounded from below.  Thus the Laplacians associated to this metric are uniformly elliptic with uniformly (in ) Hölder continuous coefficients, as we obtain estimates on half-balls exactly as before.


This concludes the required estimate near the cusp ends.  The boundary estimates further away from the cusp but near the boundary at infinity H_0 may be obtained in a manner similar to that of <cit.>.  In the neighbourhood _0, the hyperbolic metric is given by equation (<ref>).  If we introduce local coordinates at a point U on H_0, (ρ, v^i), the metric may be written

    h = dρ^2 + h_U( ρ, v )_ij dv^i dv^j/ρ^2.


We now consider p_0 = (, v^i_0) ∈∂ M_∩ C_0.  Consider ^n = _s ×^n-1_p and the euclidean reference half-ball B_+ := {(s,p^i): s^2 + | p |^2 < 1,    s ≥ 0 }, with B' denoting the half-ball with radius 1/2.  Define a map

    Ψ_p_0: B_+ Ψ_p_0(B_+)

by

    (u,v^i) = Ψ_p_0(s, p^i) = (  e^s, v_0^i +  p^i ).
 
The hyperbolic metric of (<ref>) pulls back via Ψ_p_0 to

    Ψ^*_p h = ds^2 + e^-2s h_U(  e^s, v_0 +  p )_ij dp^i dp^j.

Now the required Schauder estimates follow exactly as above.




§ LINEAR THEORY: ISOMORPHISM THEOREMS FOR Δ^H + K



In this section we state and prove an isomorphism theorem for Laplace operators of the form Δ^h + K, where K is a constant and Δ^h = ∇^* ∇ is the Laplacian with respect to a geometrically finite hyperbolic metric.  This isomorphism theorem depends on a certain asymptotic estimate for powers of the total defining function.  We then study the asymptotic estimates required to invert these Laplacians.  Finally, we state all of the conditions required to invert the specific operators given in (<ref>) that we need in the application to Einstein metrics.


To begin, suppose that either E = Σ^2_0(M) or E = M×.  We wish to prove an isomorphism theorem for

    Δ^h + K: C^k+2,α_μ(M;E) → C^k,α_μ(M;E).

We remind the reader of the exhaustion M_ of M defined in equation (<ref>), and for which the boundary Schauder estimates of <ref> apply.  The value of the weight μ∈^1+n_c will be chosen so that the following two conditions are satisfied:


  (H1) C^0_μ(M;E)∩ (Δ^h+K)= {0},

  (H2) The asymptotic estimate holds for weight μ, i.e. there exists a compact subset M_c ⋐ M and there exists δ > 0 so that in M ∖ M_c, 

    (Δ^h + K) σ^μ≥δσ^μ.



We will additionally assume that 



  (H3)  For ϵ>0 sufficiently small, the Dirichlet problem 

    (Δ^h + K) u = f,    in  M_ϵ∖∂ M_ϵ.
    
        u = 0,    on ∂ M_ϵ.
 
has a unique solution for all f∈ L^2(M_ϵ;E).

These assumptions yields the following isomorphism theorem.  

 
Let K be  constant and suppose that μ is a weight for which hypotheses (H1), (H2) and (H3) hold.  Then

    P_1 = Δ^h + K: C^k+2,α_μ(M;E) ⟶ C^k,α_μ(M;E)

is an isomorphism for all k∈_0.



Since C^k+2,α_μ(M;E)⊂ C^0_μ(M;E), we see by (H1), that the operator P_1 is injective. To show that P_1 is surjective with bounded inverse, fix f ∈ C^k,α_μ(M;E).  Now recalling the exhaustion M_ defined at the beginning of <ref>, consider for j sufficiently large the Dirichlet problem on M_1/j:

    *
        P_1 u = f,    in  M_1/j.
    
        u = 0,    on ∂ M_1/j.
 
For j sufficiently large, M_1/j are smooth manifolds with boundary, and by (H3), there is a unique solution u.  By classical elliptic PDE theory, this unique solution u is in C^k+2,α_μ(M_1/j;E). 


Moreover, by Schauder theory, the inverse map to (<ref>) is bounded, i.e. there exists C_j > 0 where

    u _k+2,α,μ; M_1/j≤ C_j  P_1 u _k,α,μ; M_1/j.



Claim: The constant C_j may be taken independently of j.  If this is not the case, then passing to a subsequence if necessary, we can suppose that there is a sequence u_j ∈ C^k+2,α_μ(M_1/j) with u_j_k+2,α,μ;M_1/j = 1 but 

    P_1 u_j _k,α,μ;M_1/j→ 0     j→∞.



Applying the boundary Schauder estimate, Proposition <ref>, we  show that for j sufficiently large the maximum value of |u_j|/σ^μ is uniformly bounded from below.  Indeed for some constant C independent of j,

    1 ≡ u_j _k+2,α,μ;M_1/j≤ C (  P_1 u_j _k,α;μ;M_1/j +  u_j _0,0,μ;M_1/j).

Thus for j sufficiently large, since P_1 u_j _0,α,μ;M_1/j→ 0 we find that 

    1/C<∼ u_j_0,0,μ;M_1/j =  u_j / σ^μ_0,0; M_1/j,

which shows that the maximum value of |u_j| / σ^μ is uniformly bounded from below in j on M_1/j.


Using the (classical) maximum principle, we will now show that the location of the maxima of |u_j|/σ^μ cannot drift to infinity.  Choose a large compact subset M_c ⋐ M so that on M ∖ M_c  the asymptotic estimate (H2) holds for weight μ.  We then assume that j is chosen so large that M_c ⋐ M_1/j.


Now consider the function |u_j| / σ^μ on M ∖ M_c∩ M_1/j.  Let p_j be a point realizing the maximum value, i.e. 

    |u_j|/σ^μ(p_j) = max_M ∖ M_c∩ M_1/j|u_j|/σ^μ.

Note that p_j ∉∂ M_1/j since u_j vanishes there.


If instead the maximum is attained in the interior of M_1/j∖ M_c, then note that the argument in <cit.> implies that

    δ|u_j|/σ^μ (p_j) ≤|P_1 u_j|/σ^μ (p_j).

To see this estimate, one simply computes as in <cit.> that for some first order operator V depending on σ,

    (Δ^h-V+δ)|u_j|/σ^μ≤(Δ^h - V + K + Δ^h σ^μ/σ^μ) |u_j|/σ^μ≤⟨(Δ^h + K) u_j,u_j⟩/σ^μ|u_j|.

Evaluating |u_j|/σ^μ at the maximum p_j and using the asymptotic estimate allows us to obtain estimate (<ref>).  


Hence we see that

    u_j _0,0,μ; M_1/j∩ M ∖ M_c = |u_j|/σ^μ (p_j) ≤1/δ|P_1 u_j|/σ^μ (p_j) ≤1/δ P_1 u_j _0,μ;M_1/j∩ M ∖ M_c≤1/δ P_1 u_j _k,α,μ;M_1/j.

Since by assumption P_1 u_j _k,α,μ;M_1/j→ 0 as j→∞, it follows that |u_j|/σ^μ (p_j) is very small when j is large.  By our earlier uniform lower bound (<ref>), this means that for j sufficiently large, the maximum value of |u_j|/σ^μ over M_1/j cannot  occur in the interior of M∖ M_c∩ M_1/j, that is, it must occur at a point q_j on the compact set M_c.


The argument now finishes in a standard way.  It is possible to choose a subsequence of q_j and of the original sequence u_j so that q_j_i→ q and u_j_i converges on compact subsets in C^k+2 to a function u∈ C^k+2,α_μ(M;E).  Additionally u ≠ 0 since (|u|/σ^μ)(q) is at least 1/c.  However, P_1 u = lim_i →∞ P_1 u_j_i, and since P_1 u_j_i_k,μ;M_1/j_i→ 0, it follows P_1 u _k,μ;M = 0 so that P_1 u = 0.  But then u = 0 by injectivity, a contradiction.  This establishes the claim and the uniform estimate 

    u _k+2,α,μ; M_1/j≤ C  P_1 u _k,α,μ; M_1/j

for a constant C>0 not depending on j.  


The remainder of the proof now finishes as in <cit.>, and we repeat the details for completeness.  Let v_j ∈ C^k+2,α_μ(M_1/j;E) be the unique solution to (<ref>) in M_1/j.  The uniform estimate (<ref>) implies 

    v_j _k+2,α,μ; M_1/j≤ C  f _k,α,μ; M_1/j≤ C  f _k,α,μ; M.

The Arzela-Ascoli theorem implies that a subsequence of v_j converges in (k+2,0,μ)-norm on M_1/j, and a diagonal subsequence argument provides a limit v ∈ C^k+2_μ(M;E) and further subsequence with v_ℓ→ v in  C^k+2_μ(M_1/j;E) for any j.  Thus (Δ^h + K) v =   f, and passing to limits in (<ref>) shows v ∈ C^k+2,α_μ(M;E) and completes the proof of both surjectivity and boundedness of the inverse.


The previous theorem can be applied to the scalar Laplacian as follows.

Given a non-negative constant K, suppose that μ is weight such that (H2) holds and C^0,0_μ(M)⊂ L^2(M).    Then

    P_1 = Δ^h + K: C^k+2,α_μ(M) ⟶ C^k,α_μ(M)

is an isomorphism.


A simple integration by parts shows that P_1 is injective on any compact domain with smooth boundary if we impose Dirichlet boundary conditions.  Since the operator is formally self-adjoint, the Fredholm alternative shows that the Dirichlet problem always has a unique solution, so condition (H3) of Theorem <ref> is satisfied.    Additionally, if u ∈ C^∞_c(M) satisfies P_1 u = 0 then by  integrating by parts again,

    0 = ∫_M u P_1 u   dv_g = ∫_M |∇ u|^2 + K u^2   dv_g,

from which it follows u = 0 since K ≥ 0.  Thus P_1 is injective on any space for which compactly supported smooth functions are dense and in particular on L^2(M).  Since we are assuming C^0_μ(M)⊂ L^2(M), this means that condition (H1) of Theorem <ref> also holds.  We can therefore apply Theorem <ref> to obtain the result.


The previous corollary extends easily to the line bundle  of smooth multiples of the hyperbolic metric since (Δ^h + K)[ u h ] = (Δ^h + K)[ u ] · h.  Note that there is no shift in weights in our conventions.

Given a non-negative constant K, suppose that μ is weight for which hypothesis (H2) holds and C^0,0_μ(M;)⊂ L^2(M;).  Then

    P_1 = Δ^h + K: C^k+2,α_μ(M;) ⟶ C^k,α_μ(M;)

is an isomorphism.


Next we consider an isomorphism theorem for the Laplacian acting on trace-free symmetric 2-tensors.  

 
Given a constant K > -n, suppose that μ is a weight for which (H2) holds and C^0_μ(M;Σ^2_0(M))⊂ L^2(M;Σ^2_0(M)).  Then

    P_2 = Δ^h +K: C^k+2,α_μ(M;Σ^2_0(M)) ⟶ C^k,α_μ(M;Σ^2_0(M))

is an isomorphism.



To apply  Theorem <ref>, we need to show that hypotheses (H1) and (H3) hold.  As we now explain, both of these follow from Koiso's trick used in slightly different settings.  To facilitate, let (N^n,h) be a hyperbolic Riemannian n-manifold taken as follows:



  * For hypothesis (H1), we take (N,h) = (M,h), which is a complete noncompact  manifold.  Note that C^∞_c(N;Σ^2_0(N)) is dense in L^2(N;Σ^2_0(N)).


  * For hypothesis (H3), we take (N,h) = (M_,h), which is a compact manifold with boundary.  Here C^∞_c(N;Σ^2_0(N)) is dense in H^2,2_0(N;Σ^2_0(N)) and encodes the Dirichlet boundary condition.


We now review Koiso's trick for the convenience of the reader <cit.>.  For (N, h) one of the manifolds above, recall that  the Lichnerowicz Laplacian of h acting on symmetric 2-tensors is given by

    Δ_L u = ∇^* ∇ u + 2 ∘ u - 2 ∘ u

where [∘ u ]_ij = 1/2 ( R_iku_j^k + R_kju_i^k) and [∘ u ]_ij = R_iklj u^kl.  For a hyperbolic metric we insert (h) = -(n-1) h and (h)_iklj = -( h_ij h_kl - h_il h_kj ) and then

    Δ_L^h u = ∇^* ∇ u -2(n-1) u + 2 (^h u) g - 2u =  ∇^* ∇ u -2 n u + 2 (^h u) h.



Suppose that λ is a constant.  In the calculations that follow we take u ∈ C^∞_c(N;Σ^2_0(N)) for either choice of N.  In both cases there are no boundary terms that arise from the integration by parts in the following calculations:

    (u, (Δ_L^h + λ)u)    = ∫_N u∇^* ∇ u + (λ-2 n) u + 2 (^h u) hh dV_h 
       = ∇ u ^2_L^2 + (λ-2n) u^2_L^2 + 2 ^h u ^2_L^2
       ≥ (λ-2n) u^2_L^2.

To improve the constant, define a 3-tensor by

    T_ijk := ∇_k u_ij - ∇_i u_jk.

In the following computation using indices, we use the convention that we sum over repeated indices and omit upper/lower position, and indices following a comma denote covariant differentiation.  Once more there are no boundary terms in the following integration by parts:

    T^2_L^2   = ∫_N T_ijk T_ijk  dV_h 
       = ∫_N (u_ij,k - u_jk,i)(u_ij,k - u_jk,i)   dV_h 
       = ∫_N 2|∇ u|^2 - 2 u_ij,k u_jk,i  dV_h 
       = ∫_N 2|∇ u|^2 + 2 u_ij,ki u_jk  dV_h 
       = ∫_N 2|∇ u|^2 + 2 (u_ij,ik + R_kiis u_sj + R_kijs u_is ) u_jk  dV_h 
       = ∫_N 2|∇ u|^2 + 2 u_ij,ik u_jk + 2R_ks u_sj u_jk + 2R_kijs u_is u_jk  dV_h 
       = ∫_N 2|∇ u|^2 - 2 u_ij,i u_jk,k -2(n-1) |u|^2 -2 (h_ijh_ks - h_kj h_is ) u_is u_jk  dV_h 
       = ∫_N 2|∇ u|^2 - 2 u_ij,i u_jk,k -2n |u|^2 + 2(^h u)^2   dV_h.

We conclude that 

    ∇ u^2_L^2 = 1/2T^2_L^2 + div^h u ^2_L^2 - ^h u^2_L^2 + n u^2_L^2.

Combining this equation with the calculation of equation (<ref>), we find that 

    (u, (Δ_L^h + λ)u)    = ∇ u ^2_L^2 + (λ-2n) u^2_L^2 + 2 ^h u ^2_L^2
       = 1/2T^2_L^2 + div^h u ^2_L^2 + ^h u^2_L^2 + (λ-n) u^2_L^2
       ≥ (λ-n) u^2_L^2.


In our analysis, we are interested in P_2 = ∇^* ∇ + K.  Note that

    (u, P_2 u)    = (u, (∇^* ∇ + K) u ) 
       = (u, Δ_L u - 2 ∘ u + 2 ∘ u +Ku) 
       = (u, Δ_L u + 2(n-1) u - 2(^h u) h + 2 u + K u) 
       = (u, Δ_L u + (2n+K) u)
       ≥ ( 2n+K - n) u^2_L^2
       ≥ ( n+K ) u^2_L^2.


Thus we obtain the estimate P_2 u_L^2≥ (n+K) u_L^2 in both of the choices for N.  Hypothesis (H1) now follows as we take values of μ so that C^k+2,α_μ embeds into L^2.  Hypothesis (H3) follows from the Fredholm alternative since solutions to the homogeneous Dirichlet problem are unique.  Thus we may apply Theorem <ref>.




 §.§ Asymptotic estimates


We now consider the asymptotic estimate (H2) above.  We separate into cases by estimating near intermediate rank cusps, near maximal rank cusps and near H_0.




  §.§.§ Intermediate rank cusps

Consider an intermediate rank cusp neighbourhood, where the rank of the cusp is f < n-1 and set  b := n-1-f.  We write the metric of (<ref>) in a slightly different way to facilitate computation.  Recalling that dϕ^2 is the round metric on ^b, let θ_0 denote the Riemannian distance from the north pole.  Thus ρ = cos(θ_0).  Then dϕ^2 = dθ_0^2 + sin^2(θ_0) d θ_α^2, where dθ_α^2 is the round metric on ^b-1.  The metric (<ref>) can then be written

    h = dr^2/r^2 cos^2(θ_0) + d θ_0^2 +  sin^2(θ_0) d θ_α^2/cos^2(θ_0) + r^2/cos^2(θ_0) dw^2,

where dw^2 is a flat metric.


Recall that for a generic metric g, the coordinate expression for the Laplacian on functions is given by 

    Δ^g u =   -1/√( | g|)∂_i ( √( | g|) g^ij∂_j u).


For the hyperbolic metric (<ref>), note that

    h = r^2(f-1)sin(θ_0)^2(b-1)/cos(θ_0)^2n(dθ_α^2) (dw^2).



A straightforward computation shows

    -Δ^h u    = cos^2(θ_0) r^2 ∂_r^2 u + (f+1) cos^2(θ_0) r ∂_r u 
       + cos^2(θ_0) ∂_θ_0^2 u + (n-2) sin(θ_0) cos(θ_0) ∂_θ_0 u + (b-1) cos^3(θ_0)/sin(θ_0)∂_θ_0 u + cos^2(θ_0)/sin^2(θ_0)Δ^^b-1 u 
        + (cosθ_0)^2/r^2Δ^dw^2 u .



Now, applying the Laplacian plus a constant K to the function u = r^μ (cosθ_0)^ν, μ, ν∈, we obtain

    (Δ^h + K)( r^μ (cosθ_0)^ν ) = [ K - (μ^2 + f μ -  b ν)cos(θ_0)^2 - ( ν(ν - (n-1) )sin(θ_0)^2  ] r^μ (cosθ_0)^ν.


Thus a simple condition ensuring that (H2) holds in this asymptotic end for some δ > 0 is that both

    K - (μ^2 + f μ -  b ν) > δ   K - ( ν(ν - (n-1) ) > δ

hold simultaneously.



  §.§.§ Maximal rank cusps
 Now consider a cusp neighbourhood of a maximal rank cusp, with metric given by (<ref>).  The reader may check that applied to u = r^μ, one obtains

    (Δ^h + K) r^μ = (K - μ(μ + (n-1)) ) r^μ,

so that (H2) holds in this asymptotic end for some δ > 0 if

    K - μ(μ + (n-1)) > 0.




  §.§.§ Asymptotic estimate near H_0
  Near a point of H_0, the metric is given by (<ref>).  A calculation shows that applied to u = ρ^ν, one obtains

    (Δ^h + K) ρ^ν = (K - ν(ν - (n-1))) ρ^ν,

so that (H2) holds near H_0 for some δ > 0 if

    K - ν(ν - (n-1)) > 0.




 §.§ Specialization to L
 In our application to Einstein metrics of the next section, we need to invert the operator L given in equation (<ref>).  In order to apply Corollaries <ref>–<ref> we must require each weight in a weight vector μ∈^1+n_c to be larger than the L^2 cutoff, as given in Lemma <ref>:

    μ_0 > n-1/2,   μ_j > -f_j/2.

where f_j is the rank of the j-th cusp.   In fact we are forced to choose μ_j > 0 in order to preserve the asymptotic class of the metric. 


In order to satisfy estimates (<ref>), (<ref>), and (<ref>) simultaneously for both K = 2(n-1) and K=-2, it suffices to choose weights for K = -2.  Unfortunately a quick analysis of (<ref>) shows that there are no positive weights for which we may obtain an isomorphism theorem.  Thus we cannot apply our method to cusps of maximal rank.


Near H_0 we must have

    μ_0 > n-1/2  μ_0 (μ_0 - (n-1) ) < -2,

which forces  

    μ_0 ∈( n-1/2, n-1/2 + √((n-1)^2-8)/2).


Near a cusp of intermediate rank, we must take μ_i slightly positive so that

    -2 > (μ_i^2 + f_i μ_i - b_i μ_0) = (μ_i^2 + f_i μ_i - (n-1-f_i) μ_0).


We now give two examples of weights for which the asymptotic estimate holds.
  If n > 4 and if (M^n,h) is a geometrically finite hyperbolic metric with n_c cusps of intermediate rank f_j, j = 1, …, n_c, then
the asymptotic estimate holds for K=-2 with weights

    μ_0 = n-2, μ_i = 1/n-2.



An easy calculation shows that μ_0 = n-2 satisfies μ_0(μ_0 - (n-1)) < -2 when n > 4, and further, μ_0 > 1/2(n-1).  Recall now that intermediate rank cusps entail that 1 ≤ f_i ≤ n-2 < n-1.  So for μ_i = 1/n-2 we have

    μ_i^2 + f_i μ_i    = 1/(n-2)^2 + f_i 1/n-2
       ≤1/(n-2)^2 + (n-2) 1/n-2
       < 2 < n 
       ≤ -2 + (n-2) 
       ≤ -2 + (n-1 -f_i)(n-2),

which gives the required estimate.


  For n=4, the asymptotic estimate for a geometrically finite hyperbolic metric (M^n,h) with n_c cusps only holds if all the cusps are of rank f_i = 1.  Any value of μ_0 ∈ (3/2,2) and μ_i sufficiently small (depending on μ_0) will satisfy the asymptotic estimate.


First, note that when n=4, the constraint of (<ref>) implies that μ_0 ∈ (3/2,2).  


Since the cusps are of intermediate rank, we find that there are only two cases to check, when f_i = 1 and when f_i =2.  When f_i = 2, we see that the constraint on μ_i reads

    μ_i^2 + 2 μ_i < μ_0 - 2,

and the right hand side is always nonpositive for admissible values of μ_0.  Thus no positive weights μ_i are admissible when f_i = 2.


When f_i = 1, we find

    μ_i^2 + μ_i < 2μ_0 - 2,

and the right hand side is now always positive for admissible values of μ_0.  Thus the constraint for μ_i can always be satisfied for μ_i sufficiently small and positive.





§ EINSTEIN METRICS NEAR A GEOMETRICALLY FINITE HYPERBOLIC METRIC



In this section we prove the existence of Einstein perturbations of a geometrically finite hyperbolic metric of Theorem <ref>.


Fix a hyperbolic metric (M,h) with n_c intermediate rank cusps  and consider the compactification of M to the manifold with corners  of <ref>.  Rescale the metric h by the square of the defining function for H_0, to obtain a partial conformal compactification

    ħ := ρ^2 h.

We then restrict to the hypersurface H_0,

    ĥ: = (ρ^2 h)|_TH_0,

obtaining a metric analogous to the conformal infinity of a conformally compact metric, except that H_0 is a noncompact manifold with boundary and ĥ is a foliated cusp metric.


We will be interested in compactly supported perturbations of ĥ, i.e. perturbations away from the cusp faces.  To this end, let U be an open set of H_0∖∂ H_0 with closure also contained in the interior of H_0, so that the inward normal exponential map of h = ρ^2 h is a diffeomorphism from a small product neighbourhood of U × [0,δ) to a collar neighbourhood of U in  away from the cusp hypersurfaces.  In what follows we implicitly use this exponential map to identify this neighbourhood with the product U × [0,δ).  If y^α are arbitrary coordinates on U, we extend them into M by declaring them to be constant along the integral curves of ∇^ħρ.  In this neighbourhood, the hyperbolic metric is of the form

    h = d ρ^2 + h_U(ρ)/ρ^2

for some family of metrics h_U on U smoothly parametrized by ρ.  To prove our main result, we first need to construct a approximate solution to the equation Q(g,t)=0.  
This is achieved by adapting the construction of asymptotic solutions to Q = 0 from Theorem 2.11 of <cit.>.  Note that we replace ∂ M with U, M with U × (0,δ), and  with U × [0,δ).


We will make use of the asymptotic expansion spaces, A^m_k,α(M) of Graham-Lee.  Recall that a function f is in A^m_k,α(U × [0,δ) ) if it can be written as a sum

    f = w_k+m + ρ w_k+m-1 + ⋯ + ρ^m w_k,

where w_j ∈ C^j,α(U × [0,δ)), and we emphasize the regularity here is taken up to the boundary ρ = 0.  A symmetric 2-tensor lies in A^m_k,α(Σ^2(U × (0,δ))) if its components in any coordinate system up to the boundary lie in A^m_k,α(U × (0,δ)).  As explained in <cit.>, these spaces are well-defined and Banach spaces under an appropriate norm.  Moreover, the gauge-adjusted Einstein operator

    Q: ρ^-2 A^m_k+2,α(Σ^2(U × (0,δ))) ×ρ^-2 A^m_k+2,α(Σ^2(U × (0,δ))) ⟶ρ^-2 A^m+2_k,α(Σ^2(U × (0,δ)))

is a smooth map.


We first describe an extension procedure that takes q̂∈ C^k,α_c(U;Σ^2(U)) of compact support on U  to a perturbation of the hyperbolic metric on M.  Choose a non-negative C^∞() bump function ψ so that ψ≡ 1 on U × [0,δ/2) and ψ is has compact support in (H_0∖∂ H_0)× [0,δ). Now extend q̂ to a tensor  on U × [0,δ) by first declaring ∇^ħρ = d ρ and then parallel translating along the ħ-geodesics normal to H_0.  Finally, for ĝ = ĥ+q̂, set 

    E(ĝ) = h + ψ.

We observe that E() = ħ.  The extension operator in coordinates essentially yields

    E() = ħ + ψ(ρ,y) (y)_αβ dy^α dy^β,

and we then define

    T: C^k,α(U;Σ^2(U))    ⟶ρ^-2 A^0_k,α(Σ^2(U × [0,δ))) 
    
    T()    = ρ^-2 E(ĝ) = h + ρ^-2ψ.

T is a smooth map of Banach spaces.  Further, the metric T() is conformally compact in U × [0,δ) since ρ^2 T() extends to a C^k,α metric on U × [0,δ).  


Now set g_1 = T().  We first check that g_1 is a first-order solution to the gauge-adjusted Einstein equation

    Q(g_1,g_1) = (g_1) + (n-1) g_1.

Note that  the gauge term vanishes when both arguments of Q are identical.
Outside of the support of ψ and in particular near any cusp face, the metric g_1 is identical to the hyperbolic metric and thus Q(g_1,g_1) = 0. Inside the support of ψ, since the metric g_1 is conformally compact and asymptotically hyperbolic, the components of Q(g_1,g_1)_jk are O(ρ^-1) relative to the coordinate system described above, and in fact Q(g_1,g_1) ∈ρ^-1 A^1_k-2,α.  Thus |Q(g_1,g_1)|_h = O( ρ ), and g_1 is a first order solution as claimed.  Unfortunately, the fact that Q(g_1,g_1) is in ρ^-1 A^1_k-2,α is insufficient to apply the isomorphism theorem since the corresponding weighted Hölder space[The precise embedding of the ρ^μ A^m_k,α spaces into our Hölder spaces is discussed at the end of this section.] will not embed into L^2. Hence the next step in the argument is to construct a finite series of metrics that solve the gauge-adjusted Einstein equation to a sufficiently high order.


Fixing g_1 as the background reference metric for Q, we now seek a higher order correction g_2 = g_1 + r such that Q(g_2,g_1)_jk = O(1)   ( =  O(ρ^0) ), or

    |Q(g_2,g_1)|_h = O( ρ^2 ).

Applying Taylor's theorem, Graham and Lee show that for an r = ρ^-1_1 ∈ C^k-1,α(U × [0,δ)),

    Q(g_1 +  r, g_1) = Q(g_1,g_1) + L( ρ^-1_1 ) + O(ρ),

where to leading order L is the operator of (<ref>).   Thus to obtain g_2 we must solve for _1 with 

    L( ρ^-1_1 ) = -ρ^-1·{ Q(g_1,g_1) }_-1,

where the notation {·}_-1 represents the coefficient of the ρ^-1 term in the Taylor expansion of Q(g_1,g_1).  One may solve for the correction tensor r above by a purely algebraic process arising through an analysis of the indicial operators of L on tensors.  Further, _1 and therefore g_2 remain compactly supported within the support of ψ since Q(g_1,g_1)_jk = O(ρ^-1) is supported within the support of ψ.  Thus the process remains completely local in the sense that Q(g_2,g_1)_jk = 0 outside the support of ψ and Q(g_2,g_1)_jk = O(1) within the support.  We thus obtain the expansion

    g_2 = g_1 + r_1 = h + ρ^-2ψ + ρ^-1_1.

Thus g_2 ∈ρ^-2 A^1_k-1,α( Σ^2(U × [0,δ) ), and
Q(g_2,q_1) ∈ρ^0 A^1_k-3,α, where we recall that we are assuming that k > n.


We may continue to improve the order of vanishing of the approximate solution inductively until the first characteristic exponent of L appears. For our dimension convention, we obtain an expansion

    g_n-1 = h + ρ^-2ψ + ρ^-1_1 + ⋯ + ρ^n-3_n-1,

where Q(g_n-1, g_1)_jk = O( ρ^n-3 ), or equivalently |Q(g_n-1,h)|_h = O(ρ^n-1).  In fact g_n-1 = ρ^-2 A^n-2_k-n+2,α(Σ^2(U × [0,δ))) and Q(g_n-1,g_1) ∈ρ^n-3 A^1_k-n,α(Σ^2(U × [0,δ))).


Finally, we summarize this discussion.  The map

    S: C^k,α( U ;Σ^2(u))    ⟶ρ^-2 A^n-2_k-n+2,α(Σ^2(U × [0,δ) ) )
    
    S()    = g_n-1.

that gives

    |Q(S(), T())|_h = O(ρ^n-1)

is a smooth map of Banach spaces.  The composition → Q( S(), T()) ∈ρ^n-3 A^1_k-n,α(Σ^2(U × [0,δ))) is additionally a smooth map.


It remains to embed the asymptotic expansion spaces into our Hölder spaces.  Recall Proposition 3.3(12) of <cit.> shows that

    A^m_k,α⊂Λ^0_k,α

continuously for any m, where Λ^0_k,α are the Graham-Lee Hölder spaces.  Consequently, 

    ρ^μ A^m_k,α⊂Λ^μ_k,α

continuously for any m.  Away from the cusp hypersurfaces our C^k,α_μ spaces are equivalent to Λ^μ_0_k,α except our conventions for measuring the norm of a symmetric 2-tensor using the metric instead of the norm of components in smooth background components lead to a shift in weight by 2 in terms of ρ.  Since the tensors above are equal to the hyperbolic metric outside the support of ψ, they have infinite order vanishing with respect to the hyperbolic metric at the cusp ends.  Thus a symmetric 2-tensor that is a compactly supported perturbation of h in U × [0,δ) lying in ρ^cΛ^m_k,α embeds continuously into C^k,α_(c+2,ν,…,ν)(M, Σ^2(M)), for any ν > 0.

 
Thus given ∈ C_c^k,α(U,Σ^2(U)) of compact support in U, there exists perturbations of the hyperbolic metric S(+) and T(+) lying in C^k-n+2,α_(0,ν,…,ν)(M,Σ^2(M)) and C^k,α_(0,ν,…,ν)(M,Σ^2(M)) respectively such that 

    Q(S(+),T(+)) ∈ C^k-n,α_(n-1,ν,…,ν)(M,Σ^2(M)),

and moreover these maps are smooth in a neighbourhood of .


Using this asymptotic solution, we can rephrase our main theorem as follows.  


For n ≥ 4, and k > n, let (M^n,h) be a geometrically finite hyperbolic metric with n_c intermediate rank cusps.  If n=4, suppose furthermore that all the cusps are of rank 1.  Let U be any open set in H_0 with closure contained in H_0∖ H_0.  Let μ=(μ_0,μ_1,…,μ_n_c) be the multi-weight given by 

    μ_0=n-2, μ_i=1/n-2

if n>4, and otherwise be a multi-weight as specified by Proposition <ref> if n=4.
Then there exists > 0 such that for any smooth symmetric 2-tensor  compactly supported in  U and perturbation ĝ = ĥ + with _k,α <, there is an asymptotically hyperbolic metric g=S(ĥ+)+r on M with r∈ C^k-n,α_μ(M;Σ^2(M)) such that 
ρ^2 g is continuous on U and (ρ^2 g)|_U = ĝ and  g is Einstein, i.e. 

    (g) + (n-1)g = 0.



Before starting the proof, we discuss the strategy.  Given a boundary metric  and the approximate solutions described above, we will use the inverse function theorem in order to add a “correction" tensor r that yields an exact solution to the gauge-adjusted Einstein equation.  In order to apply the isomorphism results of Propositions <ref> and  <ref>, we require the correction term to lie in Hölder spaces that embed into L^2.  This means the weight in the boundary defining function ρ for H_0 must be at least (n-1)/2.  It is possible to construct an approximate solution with this weight in ρ using roughly C^n/2 control of the boundary norm of , and we leave precise details to the interested reader.  Instead, for simplicity, we fix the weight at H_0 to be μ_0 = n-2, which requires C^k control of  with k > n as described in the asymptotic solution above.
 

The remainder of this section is occupied with the proof of the main theorem.






 §.§ The inverse function theorem argument


For some μ to be specified, define an open subset  of the Banach space ×  C^k-n+2,α_μ(M; Σ^2(M)) by

    := { (t,r): +t, S(+t),S(+t)+r  }

and consider the map

    :    ⟶×  C^k-n,α_μ(M; Σ^2(M)), 
    ( t, r )    = (   t, Q(   S(  +t ) + r,  T( +t )  )   ).


By equation (<ref>), the corresponding discussion in <cit.> adapted to our notation and Corollary <ref>, this is a smooth map with the property that (0,0) = (0,0) and the linearization at this point is

    D_(0,0) ( τ, r )    = ( τ,D_1 Q_(h,h) ( τ DS_ĥq̂ + r) + D_2 Q_(h,h)( τ DT_ĥq̂) ) 
       = (p̂, L r + τ J q̂ ),

with L defined in equation (<ref>) and J q̂ =  D_1 Q_(h,h) DS_ĥq̂  + D_2 Q_(h,h)DT_ĥq̂, exactly as in Graham-Lee.  Thus a unique solution to 

    D_(ĥ,0) (q̂,r) = (ŵ,v)

is given by q̂ = ŵ and r = L^-1(v-Jŵ), provided we can invert L.


Using Propositions <ref> and  <ref>  and our choice of  multi-weight μ, our isomorphism theorem provides a bounded inverse for L between weighted Hölder spaces.  Thus if ĝ is sufficiently close to ĥ, i.e. if  is sufficiently small in (k-n,α)-norm and |t|<1, there is a family of metrics g_t for t∈ (-1,1) of the form g_t = S(+t) + r_t, with t↦ r_t ∈ C^k-n+2,α_μ(M;Σ^2(M)), such that Q( g_t, T(+t) ) = 0, by the inverse function theorem.  Note that by construction ρ^2 g_t|_U = + t since the tensor ρ^2 r_t ∈ρ^2 C^k-n+2,α_μ(M;Σ^2(M)) vanishes on H_0, and the choice of weight vector ensures that the asymptotic structure of the metric is preserved.


Note that the metric g is smooth in M by interior elliptic regularity.




 §.§ Returning to the Einstein equation


The final step in the proof is to argue that g_t = S(+t) + r_t is an Einstein metric.  By shrinking the neighbourhood obtained from the inverse function theorem, we can always ensure that g can be made as close to the hyperbolic metric h as desired in C^k-n,α_μ-norm, and as a consequence (g) < 0.


Set τ = T(+t).  Recalling that Q(g_t,τ) = 0 and that the leading part of S(+t) is given by T(+t), we find that g_t = τ + r_t', for t↦ r_t' ∈ C^k-n+2,α_δ(M;Σ^2(M)), where δ = (μ_0 + 1, μ_1, …, μ_n_c ), in other words, r_t' has faster decay at the H_0 hypersurface.


Let ω_t be the gauge-breaking DeTurck vector field

    ω_t = g_t τ^-1δ_g_t G_g_tτ.

Writing τ = g_t - r_t', and using the mapping properties of these operators shows that ω_t ∈ C^k-1,α_δ(M;Σ^2(M)).  Thus |ω|^2_g = O(σ^2 δ), i.e. ω vanishes in each end of M.  Applying the Bianchi operator δ_g_t G_g_t to Q(g_t,τ) = 0 and commuting derivatives shows that |ω_t|^2_g satisfies a differential inequality in M:

    Δ |ω_t|^2_g ≤ 2 η |ω_t|_g^2,

where η is a negative constant that comes from the hypothesis on Ricci curvature.  As ω_t → 0 in each end, it follows that ω_t = 0 by the classical maximum principle.  Thus

    0 = Q(g_t,τ) = (g_t) + (n-1)g_t + δ_g_t^* ω_t = (g_t) + (n-1) g_t,

and g_t is Einstein.


This concludes the proof of Theorem <ref>.
amsalpha

