Geometric inequalities for critical metrics of the volume functional

Abstract

The goal of this article is to investigate the geometry of critical metrics of the volume functional on an n-dimensional compact manifold with (possibly disconnected) boundary. We establish sharp estimates to the mean curvature and area of the boundary components of critical metrics of the volume functional on a compact manifold. In addition, localized version estimates to the mean curvature and area of the boundary of critical metrics are also obtained.

On the Uniqueness of Schwarzschild–de Sitter Spacetime

“…2 and 3 are not exclusive of the static realm, but they hold more generally for large classes of real-valued functions. In fact, other recent applications of these same properties have appeared in [5], where they have been used to study critical metrics of the volume functional, and in [2] where the classical torsion problem is discussed. From Section 4 we start focusing exclusively on static solutions.…”