Rigidity in the Positive Mass Theorem with $C^0$ Decay

Abstract

Let $g$ be a smooth metric on ${\mathbb{R}}^3$ with non-negative scalar curvature. We show that if $g$ satisfies $|g(x)-g_{\text{euc}}(x)|=O(|x{|}^{-1-\ddot{I}„})$ for some $\ddot{I}„>0$ then $g$ must be flat.

This work is now superseded by arXiv:2606.19123. 17 pages, comments are welcome!