Singular positive mass theorem with arbitrary ends

TL;DR

This paper advances the positive mass theorem to include asymptotically flat manifolds with arbitrary $C^0$ ends and metrics with limited regularity, broadening the scope of geometric analysis.

Contribution

It establishes the positive mass theorem for manifolds with $C^0$ ends and $W^{1,p}_{loc}$ metrics, introducing new techniques for handling non-compact singular sets.

Findings

Proved positive mass theorem under new regularity conditions.

Developed methods to address non-compactness of singular sets.

Extended the theorem to manifolds with arbitrary ends.

Abstract

Motivated by the recent progress on positive mass theorem for asymptotically flat manifolds with arbitrary ends and the Gromov's definition of scalar curvature lower bound for continuous metrics, we start a program on the positive mass theorem for asymptotically flat manifolds with $C^0$ arbitrary ends. In this work as the first step, we establish the positive mass theorem of asymptotically flat manifolds with $C^0$ arbitrary ends when the metric is $W^{1,p}_{\mathrm{loc}}$ for some $p\in(n,\infty]$ and is smooth away from a non-compact closed subset with Hausdorff dimension $n-\frac{p}{p-1}$. New techniques are developed to deal with non-compactness of the singular set.