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 $\vert g(x)-g_{\text{euc}}(x)\vert = O(\vert x\vert^{-1-τ})$ for some $τ> 0$ then $g$ must be flat.