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

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