Positive Mass Theorem on Manifolds admitting Corners along a Hypersurface

TL;DR

This paper extends the positive mass theorem to certain non-smooth asymptotically flat manifolds with corners along a hypersurface, using an approximation scheme and boundary conditions.

Contribution

It introduces an approximation method for mollifying metrics with corners and proves the positive mass theorem under these non-smooth conditions.

Findings

Positive mass theorem holds for manifolds with corners under specific boundary conditions

An approximation scheme effectively mollifies non-smooth metrics

The geometric boundary condition is crucial for the theorem's validity

Abstract

We study a class of non-smooth asymptotically flat manifolds on which metrics fails to be $C^1$ across a hypersurface $\Sigma$. We first give an approximation scheme to mollify the metric, then we prove that the Positive Mass Theorem still holds on these manifolds if a geometric boundary condition is satisfied by metrics separated by $\Sigma$.