Rigidity for critical metrics of the volume functional

TL;DR

This paper establishes rigidity results for Miao-Tam critical metrics and static metrics with positive scalar curvature, showing geodesic balls maximize boundary volume under certain conditions and extending classical rigidity theorems.

Contribution

It proves that geodesic balls are volume maximizers among Miao-Tam critical metrics with Einstein boundary and extends rigidity theorems to higher-dimensional static metrics.

Findings

Geodesic balls maximize boundary volume among Miao-Tam critical metrics with Einstein boundary.

Extension of classical rigidity theorems to n-dimensional static metrics with positive scalar curvature.

Provides partial answers to the Cosmic no-hair conjecture.

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}.