Riemannian Penrose inequality without horizon in dimension three

Abstract

Based on the $\mu$-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 ADM mass $m$ at the infinity of ${\mathbb{R}}^3$ satisfies $m\ge \sqrt{\frac{A_g}{16\pi }}$, where $A_g$ is denoted to be the area infimum of embedded closed surfaces homologous to ${\mathbb{S}}^2(1)$ in ${\mathbb{R}}^3\setminus \{O\}$. 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 $\sqrt{\frac{A_g}{16\pi }}$.

16 pages