Some stability results of positive mass theorem for uniformly asymptotically flat $3$-manifolds

Abstract

In this paper, we show that for a sequence of orientable complete uniformly asymptotically flat $3$-manifolds $(M_i,g_i)$ with nonnegative scalar curvature and ADM mass $m(g_i)$ tending to zero, by subtracting some open subsets $Z_i$, whose boundary area satisfies $\mathrm{Area}(\partial Z_i)\le Cm(g_i{)}^{1/2-\epsilon }$, for any base point $p_i\in M_i\setminus Z_i$, $(M_i\setminus Z_i,g_i,p_i)$ converges to the Euclidean space $({\mathbb{R}}^3,g_E,0)$ in the $C^0$ modulo negligible volume sense. Moreover, if we assume that the Ricci curvature is uniformly bounded from below, then $(M_i,g_i,p_i)$ converges to $({\mathbb{R}}^3,g_E,0)$ in the pointed Gromov-Hausdorff topology.

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