Gromov's Euclidean Endpoint $C^0$ Rigidity for the Positive Mass Theorem

Abstract

We prove Gromov's Euclidean endpoint $C^0$ rigidity conjecture. Let $g$ be a smooth complete metric on $\R^3$ with non-negative scalar curvature. If $$ |g-g_{\Euc}|=o(r^{-1}),\qquad r=|x|\to\infty, $$ then $(\R^3,g)$ is isometric to Euclidean space.