The equality case of the Penrose inequality for asymptotically flat graphs

TL;DR

This paper proves that the equality case of the Penrose inequality for asymptotically flat hypersurfaces in Euclidean space characterizes Schwarzschild solutions, extending previous results to all dimensions.

Contribution

It establishes that equality in the Penrose inequality implies the hypersurface is a Schwarzschild solution, generalizing prior work to all dimensions.

Findings

Equality in Penrose inequality characterizes Schwarzschild solutions.

Asymptotically flat hypersurfaces with minimal boundary are mean convex.

The proof involves ellipticity and maximum principles for scalar curvature.

Abstract

We prove the equality case of the Penrose inequality in all dimensions for asymptotically flat hypersurfaces. It was recently proven by G. Lam that the Penrose inequality holds for asymptotically flat graphical hypersurfaces in Euclidean space with non-negative scalar curvature and with a minimal boundary. Our main theorem states that if the equality holds, then the hypersurface is a Schwarzschild solution. As part of our proof, we show that asymptotically flat graphical hypersurfaces with a minimal boundary and non-negative scalar curvature must be mean convex, using the argument that we developed earlier. This enables us to obtain the ellipticity for the linearized scalar curvature operator and to establish the strong maximum principles for the scalar curvature equation.