A positive mass theorem for Lipschitz metrics with small singular sets

TL;DR

This paper extends the positive mass theorem to Lipschitz metrics with low-dimensional singular sets, showing nonnegativity of mass under minimal regularity and singularity conditions, especially in dimensions less than 8 or on spin manifolds.

Contribution

It proves the positive mass theorem for Lipschitz metrics with singular sets of Minkowski dimension less than half the manifold dimension, broadening the class of metrics where the theorem applies.

Findings

Mass is nonnegative for Lipschitz metrics with small singular sets.

Results hold in dimensions less than 8 or on spin manifolds.

Conjecture that singular set dimension can be less than n-1.

Abstract

We prove that the positive mass theorem applies to Lipschitz metrics as long as the singular set is low-dimensional, with no other conditions on the singular set. More precisely, let $g$ be an asymptotically flat Lipschitz metric on a smooth manifold $M^n$, such that $n<8$ or $M$ is spin. As long as $g$ has bounded $C^2$ norm and nonnegative scalar curvature on the complement of some singular set $S$ of Minkowski dimension less than $n/2$, the mass of $g$ must be nonnegative. We conjecture that the dimension of $S$ need only be less than $n-1$ for the result to hold. These results complement and contrast with earlier results of H. Bray, P. Miao, and Y. Shi and L.-F. Tam, where $S$ is a hypersurface.