A Minkowski inequality for hypersurfaces in the Anti-deSitter-Schwarzschild manifold

TL;DR

This paper establishes a sharp Minkowski inequality for hypersurfaces in the Anti-deSitter-Schwarzschild manifold, extending classical results and connecting to the Penrose inequality through inverse mean curvature flow techniques.

Contribution

It introduces a new monotonicity formula for inverse mean curvature flow in higher dimensions, generalizing Minkowski inequalities to the Anti-deSitter-Schwarzschild setting.

Findings

Proves a sharp inequality for hypersurfaces in Anti-deSitter-Schwarzschild space.

Extends classical Minkowski inequality to higher dimensions.

Provides a geometric interpretation related to the Penrose inequality.

Abstract

We prove a sharp inequality for hypersurfaces in the n-dimensional Anti-deSitter-Schwarzschild manifold for general n greater or equal to 3. This inequality generalizes the classical Minkowski inequality for surfaces in the three dimensional Euclidean space, and has a natural interpretation in terms of the Penrose inequality for collapsing null shells of dust. The proof relies on a new monotonicity formula for inverse mean curvature flow, and uses a geometric inequality established by the first author in [3].