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

TL;DR

This paper proves that sequences of asymptotically flat 3-manifolds with nonnegative scalar curvature and vanishing ADM mass converge to Euclidean space under certain conditions, with stability results depending on curvature bounds.

Contribution

It establishes stability results for the positive mass theorem in asymptotically flat 3-manifolds, including convergence in $C^0$ and Gromov-Hausdorff senses, with boundary area control.

Findings

Sequences with ADM mass tending to zero converge to Euclidean space.

Convergence in $C^0$ sense modulo negligible volume under boundary area conditions.

Gromov-Hausdorff convergence when Ricci curvature is bounded below.

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 $\mathrm{Area}(\partial Z_i) \leq Cm(g_i)^{1/2 - \varepsilon}$, 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.