Rigidity of Riemannian Penrose inequality with corners and its implications

TL;DR

This paper proves that singular metrics achieving equality in the Riemannian Penrose inequality must be smooth in certain coordinates, leading to rigidity results for hypersurfaces in Schwarzschild manifolds.

Contribution

It establishes the smoothness and rigidity of metrics and hypersurfaces related to the Riemannian Penrose inequality, extending understanding of equality cases.

Findings

Singular metrics attaining the optimal inequality are necessarily smooth in specific coordinates.

Rigidity of isometric hypersurfaces with the same mean curvature in Schwarzschild manifolds.

Implications for the structure of solutions achieving equality in the Penrose inequality.

Abstract

Motivated by the rigidity case in the localized Riemannian Penrose inequality, we show that suitable singular metrics attaining the optimal value in the Riemannian Penrose inequality is necessarily smooth in properly specified coordinates. If applied to hypersurfaces enclosing the horizon in a spatial Schwarzschild manifold, the result gives the rigidity of isometric hypersurfaces with the same mean curvature.