The lock principle for scalar curvature

TL;DR

This paper establishes a positive mass theorem for certain singular spin manifolds, allowing some singularities to be mean-concave if others are sufficiently mean-convex, using a novel transfer of convexity defects.

Contribution

It introduces a new approach to handle hypersurface singularities in positive mass theorems by transferring convexity defects between singularity components.

Findings

Proves positive mass theorem for asymptotically flat spin manifolds with hypersurface singularities.

Allows some singular components to be mean-concave under specific conditions.

Uses initial data sets with a second fundamental form to transfer convexity defects.

Abstract

We prove a Riemannian positive mass theorem for asymptotically flat spin manifolds with hypersurface singularities. Unlike earlier results, some components of the singular set may be mean-concave, provided that other components of the singular set are sufficiently mean-convex. Our proof uses initial data sets where a suitably chosen second fundamental form transfers convexity defects between different singularity components.