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

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