Ricci curvature rigidity for weakly asymptotically hyperbolic manifolds

Abstract

A rigidity result for weakly asymptotically hyperbolic manifolds with lower bounds on Ricci curvature is proved without assuming that the manifolds are spin. The argument makes use of a quasi-local mass characterization of Euclidean balls from [9] [14] and eigenfunction compactification ideas from [12].

Some Geometric Problems of Conformally Compact Einstein Manifolds

“…for some > 0, where hg = Ricg + (n − 1)g. It is natural, at least in some cases, to expect that the solution g(t) to (2) should stay in a small neighborhood of g0 if the initial metric g0 satisfies (14). To verify this, we need the estimate of h(t) = h g(t) .…”

Rigidity of conformally compact manifolds with the round sphere as the conformal infinity

“…We require a slightly stronger decay rate. Theorem 1.3 is related to a theorem of Bonini, Miao, and Qing [3] where condition (ii) on the scalar curvature in Theorem 1.3 is replaced by a restriction on dimension, namely 2 n 6. 1 Note that if g is ALH of order α > 2 or under the weaker assumption |K(x) + 1| = O(φ(t)e −2t ) with φ ∈ L 1 (R + ), we have |R + n(n + 1)| = o(e −2t ).…”

A Positive Mass Theorem on Asymptotically Hyperbolic Manifolds with Corners along a Hypersurface

“…Instead of solving an equation which is a perturbation of Laplace equation as in [13,18] for asymptotically flat case, we realize, with our experience in [5,16], that we should consider an equation which is a perturbation of the eigenfunction equation…”

“…As in [5], we define weakly asymptotically hyperbolic manifolds as follows: Definition 2.1. A connected complete Riemannian manifold (M n , g) is said to be weakly asymptotically hyperbolic of class C m,β if g is conformally compact of class C m,β with m + β ≥ 2 and |dρ| 2 g = 1 on ∂M for a defining function ρ.…”