Ricci curvature rigidity for weakly asymptotically hyperbolic manifolds (Q851506)

The authors prove a rigidity result for weakly asymptotically hyperbolic manifolds with lower bounds on Ricci curvature, using a mixture of ideas developed in previous work of the authors [Miao, Adv. Theor. Math. Phys. 6, No. 6, 1163--1182 (2002) and Qing, Int. Math. Res. Not. 2003, No. 21, 1141--1153 (2003; Zbl 1042.53031)], and by \textit{Y. Shi} and \textit{L.-F. Tam} [see J. Differ. Geom. 62, No. 1, 79--125 (2002; Zbl 1071.53018)]). It is important to remark that, in dimensions \(2\leq n \leq 6\), the proof is made without assumption that the manifold is Spin. The main result is: Let \((X^{n+1},g)\) be a weakly asymptotically hyperbolic manifold of order \(C^{3, \alpha}\). Assume that \((X,g)\) has the standard round sphere \((S^{n},[h_{0}])\) as its conformal infinity and satisfies Ric\((g) \geq -ng\). Let \(r\) be the special defining function such that \(g=\frac{1}{\sinh^{2}r}\{dr^{2}+g_{r}\}\) in a neighborhood of \(\partial X\) and \(g_{0}=h_{0}\). Then, if \(2\leq n \leq 6\) and \(\text{Tr} \left ( \frac{d}{dr}g_{r} \right ) \in \Lambda _{0,\beta}^{s}(X)\) for some \(s > 1\), \((X,g)\) is isometric to the hyperbolic space \(\mathbb{H}^{n+1}\).