Rigidity for critical metrics of the volume functional

Abstract

Geodesic balls in a simply connected space forms $\mathbb{S}^n$, $\mathbb{R}^{n}$ or $\mathbb{H}^{n}$ are distinguished manifolds for comparison in bounded Riemannian geometry. In this paper we show that they have the maximum possible boundary volume among Miao-Tam critical metrics with connected boundary provided that the boundary of the manifold is an Einstein hypersurface. In the same spirit we also extend a rigidity theorem due to Boucher et al. \cite{Bou} and Shen \cite{Shen} to $n$-dimensional static metrics with positive constant scalar curvature, which provides another proof of a partial answer to the Cosmic no-hair conjecture previously obtained by Chru\'sciel \cite{Chrus}.