Regularity and rigidity of asymptotically hyperbolic manifolds

Abstract

In this paper, we study some intrinsic characterization of conformally compact manifolds. We show that, if a complete Riemannian manifold admits an essential set and its curvature tends to −1 at infinity in certain rate, then it is conformally compactifiable and the compactified metrics can enjoy some regularity at infinity. As consequences we prove some rigidity theorems for complete manifolds whose curvature tends to the hyperbolic one in a rate greater than 2.

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

“…These coordinates enjoy a nice property we shall exploit in Section 5: we remark that they satisfy a certain Neumann condition at infinity. A similar idea was already present in an implicit form in [27,Section 3], where they construct even harmonic charts (with respect to the metric g) on the double of M. In all that follows we select a pointp 0 ∈ M and modified Fermi coordinate charts x in a neighborhood U ofp 0 as we constructed in the previous section such that x p 0 = 0 so that = t −1 x form a coordinate system in a neighborhood ofp 0 in M. Up to a linear redefinition of the coordinates x , we can assume that g p 0 = .…”

“…This idea was also employed in [27]. where H = n + O e −as is the mean curvature of the level sets of s (see Proposition 2.9).…”

“…Results in this direction were first proved by Hu et al in their nice article [27]. Their result are divided in two parts:…”

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.…”

Low regularity Poincaré–Einstein metrics

“…The first author [3] showed that if (M, g) is a noncompact Riemannian manifold with an essential subset K and sectional curvatures approaching −1 to order e −ωr with ω > 1, together with similar decay on the covariant derivative of the curvature, then g admits a C 0,1 conformal compactification. For related results, see [4,5,7,9].…”