Compact approach to the positivity of Brown-York mass and rigidity of manifolds with mean-convex boundaries in flat and spherical contexts

TL;DR

This paper presents a compact spinorial proof of the positivity of Brown-York mass and establishes rigidity results for manifolds with mean-convex boundaries in flat and spherical geometries, extending previous results and conjectures.

Contribution

It introduces a novel, compact spinorial approach to prove positivity and rigidity results, including new spherical case results and a version of Min-Oo's conjecture.

Findings

Proves positivity of Brown-York mass without non-smooth hypersurfaces

Provides optimal eigenvalue bounds for Dirac operators on mean-convex boundaries

Establishes rigidity results for manifolds with mean-convex boundaries in flat and spherical spaces

Abstract

In this article we develope a spinorial proof of the Shi-Tam theorem for the positivity of the Brown-York mass without necessity of building non smooth infinite asymptotically flat hypersurfaces in the Euclidean space and use the positivity of the ADM mass proved by Schoen-Yau and Witten. This same compact approach provides an optimal lower bound \cite{HMZ} for the first non null eigenvalue of the Dirac operator of a mean convex boundary for a compact spin manifold with non negative scalar curvature, an a rigidity result for mean-convex bodies in flat spaces. The same machinery provides analogous, but new, results of this type, as far as we know, in spherical contexts, including a version of Min-Oo's conjecture.