A Positive Mass Theorem for Continuous Metrics

Abstract

Let $g$ be a continuous metric on ${\mathbb{R}}^3$ which is asymptotically flat in the sense that $|g_{ij}(x)-\hat{I}{´}_{ij}|=O(|x{|}^{-\ddot{I}„})$ for some $\ddot{I}„>\frac{1}{2}$. Further assume that $g$ can be uniformly approximated on compact sets by smooth metrics with almost non-negative scalar curvature. For such a metric $g$, we define a synthetic ADM mass $m(g)$ using harmonic functions. The harmonic mass $m(g)$ coincides with the usual ADM mass whenever $g$ is smooth and decays rapidly enough that the latter is defined. The harmonic mass can also be computed as a limit of the $C^0$ local mass introduced by Burkhardt-Guim. Our main result is a positive mass theorem: the harmonic mass satisfies $m(g)\ge 0$ and if $m(g)=0$ then $g$ is flat.

This supersedes arXiv:2605.29915. 35 pages, comments are welcome!