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

Conghan Dong

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.

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 -manifolds with nonnegative scalar curvature and ADM mass tending to zero, by subtracting some open subsets , whose boundary area satisfies , for any base point , converges to the Euclidean space in the modulo negligible volume sense. Moreover, if we assume that the Ricci curvature is uniformly bounded from below, then converges to in the pointed Gromov-Hausdorff topology.
Paper Structure (7 sections, 29 theorems, 132 equations)

This paper contains 7 sections, 29 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Asymptotically flat $3$-manifolds
  4. Space with Ricci curvature bounded from below
  5. Effective estimates on outer region
  6. $C^0$ modulo negligible volume convergence
  7. Removable singularity under Ricci lower bound

Key Result

Theorem 1.1

Let $(M^3, g)$ be an AF $3$-manifold with nonnegative scalar curvature. Then $m(g) \geq 0$, and equality holds if and only if $(M,g)=(\mathbb{R}^3, g_E)$ isometrically.

Theorems & Definitions (54)

  • Theorem 1.1
  • Conjecture 1.3
  • Remark 1.1
  • Definition 1.1
  • Theorem 1.4
  • Remark 1.2
  • Theorem 1.5
  • Theorem 2.1
  • Theorem 2.2: Theorem 1.2 in BKKS22
  • Proposition 2.3
  • ...and 44 more