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 $m\geq \sqrt{\frac{A_g}{16\pi}}$, where $A_g$ is denoted to be the area infimum of embedded closed surfaces homologous to $\mathbb S^2(1)$ in $\mathbb R^3\setminus{O}$. 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 $\sqrt{\frac{A_g}{16\pi}}$.