Conformal Compactification of Asymptotically Locally Hyperbolic Metrics II: Weakly ALH Metrics

Abstract

In this paper we pursue the work initiated in [6,7]: study the extent to which conformally compact asymptotically hyperbolic metrics can be characterized intrinsically. We show how the decay rate of the sectional curvature to −1 controls the Hölder regularity of the compactified metric. To this end, we construct harmonic coordinates that satisfy some Neumann-type condition at infinity. Combined with a new integration argument, this permits us to recover to a large extent our previous result without any decay assumption on the covariant derivatives of the Riemann tensor.

Weakly asymptotically hyperbolic manifolds

“…In the first part of our work here we show several properties of weakly asymptotically hyperbolic metrics, followed by some results that highlight the importance of the extrinsic curvature of the boundary, and which are in some sense complementary to those of [5] and [13]. Under a slightly stronger regularity assumption, which implies that the metric is C 1,1 conformally compact but not necessarily C 2 , we introduce a conformally invariant tensor that agrees with the trace-free extrinsic curvature along the boundary.…”

“…Remark A. 13. As is evident from the proofs of Lemma A.10 and Proposition A.11, the presence of logarithms in expansions of solutions to I(P)u = f is a consequence of the algebraic structure of P, and the exponents appearing in the expansion of f .…”

“…These studies have all required that the metric be conformally compact of at least class C 2 ; indeed in these works the notion of asymptotic hyperbolicity is defined in terms of conformal compactification. However, it is not clear whether a complete manifold with asymptotically negative curvature necessarily admits such a compactification; to our knowledge, the best results available are those of [13] (see also [7], [14]), where it is shown that if the sectional curvatures of a complete manifold approach −1 to second order at infinity then the manifold is C 1,β conformally compact for every β ∈ (0, 1). (In fact, the work [6] presents an example of a manifold for which the curvature operator approaches the negative identity operator to first order, but for which no Lipschitz conformal compactification exists.)…”

Low regularity Poincaré–Einstein metrics

“…In [4], the first author and Romain Gicquaud addressed the special case of Einstein metrics, and showed that every Poincaré-Einstein manifold with an essential subset and QHCD has a C 1,α compactification for any α ∈ (0, 1). On the other hand, in a subsequent paper [5], Gicquaud remarked that it does not seem unreasonable to believe that there exist Poincaré-Einstein metrics with QHCD that have no C 2 conformal compactification.…”

Some Geometric Problems of Conformally Compact Einstein Manifolds

“…Fortunately, we do have intrinsic definition for conformally compact manifolds, for details, please see 24 and. 30 Secondly, for a C k,μ , k ≥ 0, conformally compact manifold, the defining function may not be unique. However, for any two defining functions φ1 and φ2, letḡ1 = φ 2 1 g,ḡ2 = φ 2 2 g, then we haveḡ…”