Riemannian-Penrose inequality without horizon in dimension three

Abstract:

Based on Gromov’s $\mu$-bubble method we are able to prove the following version of Riemannian-Penrose inequality without horizon: if $g$ is a complete metric on $\mathbb R^3\setminus \{O\}$ with nonnegative scalar curvature, which is asymptotically flat around the infinity of $\mathbb R^3$, then the Arnowitt-Deser-Misner mass $m$ at the infinity of $\mathbb R^3$ satisfies $m\geq \sqrt {\frac {A_g}{16\pi }}$, where $A_g$ is denoted to be the area infimum of embedded closed surfaces homologous to $\mathbb S^2(1)$ in $\mathbb R^3\setminus \{O\}$. Note that this infimum need not be achieved, for example in the case of a horn with an asymptotically cylindrical end. Moreover, the equality holds if and only if there is a strictly outer-minimizing minimal $2$-sphere such that the region outside is isometric to the half Schwarzschild manifold with mass $\sqrt {\frac {A_g}{16\pi }}$.