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 \(\textrm{Area}(\partial Z_i) \le C m(g_i)^{\frac{1}{2}- \varepsilon }\), 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.