On a Localized Riemannian Penrose Inequality

TL;DR

This paper establishes a localized Riemannian Penrose inequality relating horizon area to boundary area and mean curvature in 3D manifolds with nonnegative scalar curvature, relevant to black hole horizons in general relativity.

Contribution

It introduces a new inequality connecting horizon area with boundary area and mean curvature, extending the classical Penrose inequality to localized settings.

Findings

Derived a new inequality for 3D manifolds with boundary and nonnegative scalar curvature.

Linked the inequality to the properties of black hole horizons in general relativity.

Provides a mathematical framework for analyzing apparent horizons in spacetime slices.

Abstract

Consider a compact, orientable, three dimensional Riemannian manifold with boundary with nonnegative scalar curvature. Suppose its boundary is the disjoint union of two pieces: the horizon boundary and the outer boundary, where the horizon boundary consists of the unique closed minimal surfaces in the manifold and the outer boundary is metrically a round sphere. We obtain an inequality relating the area of the horizon boundary to the area and the total mean curvature of the outer boundary. Such a manifold may be thought as a region, surrounding the outermost apparent horizons of black holes, in a time-symmetric slice of a space-time in the context of general relativity. The inequality we establish has close ties with the Riemannian Penrose Inequality, proved by Huisken and Ilmanen, and by Bray.