Stability of the positive mass theorem in dimension three

Conghan Dong

Published 2022-11-12Version 1

In this paper, we show that for a sequence of oriented complete pointed uniformly asymptotically Euclidean $3$-manifolds $(M_i, g_i, p_i)$ with non-negative integrable scalar curvature $R_{g_i}\geq 0$, if their mass $m(g_i)\to 0$, then by subtracting some subsets $Z_i\subset M_i$ whose boundary area $|\partial Z_i|\leq C m(g_i)^{1/2}$, up to diffeomorphisms, $(M_i \setminus Z_i, g_i, p_i)$ converge to the Euclidean space $\mathbb{R}^3$ in the pointed metric topology. This confirms Huisken-Ilmanen's conjecture in terms of the flat metric topology. Moreover, if we assume the Ricci curvature bounded from below uniformly by $Ric_{g_i}\geq -2 \Lambda$, then $(M_i, g_i, p_i)$ converge to $\mathbb{R}^3$ in the pointed Gromov-Hausdorff topology.