Ricci curvature rigidity for weakly asymptotically hyperbolic manifolds

Abstract

Rigidity questions for asymptotically hyperbolic manifolds have been studied by many authors under various assumptions. In [10], Min–Oo proved a scalar curvature rigidity theorem for manifolds which are spin and are asymptotic to the hyperbolic space in a strong sense. In [1], Andersson and Dahl improved the scalar curvature rigidity for asymptotically locally hyperbolic spin manifolds. They also established the rigidity for conformally compact Einstein manifolds with spin structure. More recent related works are in [4], [15] and [16]. It is interesting to ask whether the spin structure is necessary to assure the rigidity. In [7], Listing was able to obtain a nonspin rigidity at the expense of replacing scalar curvature bound by sectional curvature bound. Very recently, in [12], Qing established the rigidity for conformally compact Einstein manifolds of dimension less than 7 without assuming spin structure. The proof in [12] uses conformal compactifications by positive eigenfunctions and the classic positive mass theorem proved by Schoen and Yau [13] for asymptotically flat manifolds. Based on ideas in [12] combined with a quasi-local mass characterization of Euclidean balls in [9], in this paper, we prove a Ricci curvature rigidity theorem for weakly asymptotically hyperbolic manifolds.