Stability of Euclidean 3-space for the positive mass theorem

TL;DR

This paper proves the stability of Euclidean space under the positive mass theorem for asymptotically flat 3-manifolds with nonnegative scalar curvature, confirming a conjecture and establishing bounds on boundary areas.

Contribution

It demonstrates the stability of Euclidean space in the positive mass theorem context and confirms Huisken and Ilmanen's conjecture, with quantitative bounds on boundary areas.

Findings

Convergence of manifolds to Euclidean space in Gromov-Hausdorff topology

Confirmation of Huisken and Ilmanen's conjecture

An almost quadratic upper bound for boundary area

Abstract

We show that the Euclidean 3-space $\mathbb{R}^3$ is stable for the Positive Mass Theorem in the following sense. Let $(M_i,g_i)$ be a sequence of complete asymptotically flat $3$-manifolds with nonnegative scalar curvature and suppose that the ADM mass $m(g_i)$ of one end of $M_i$ converges to $0$. Then for all $i$, there is a subset $Z_i$ in $M_i$ such that $M_i\setminus Z_i$ contains the given end, the area of the boundary $\partial Z_i$ converges to zero, and $(M_i\setminus Z_i,g_i)$ converges to $\mathbb{R}^3$ in the pointed measured Gromov-Hausdorff topology for any choice of basepoints. This confirms a conjecture of G. Huisken and T. Ilmanen. Additionally, we find an almost quadratic upper bound for the area of $\partial Z_i$ in terms of $m(g_i)$. As an application of the main result, we also prove R. Bartnik's strict positivity conjecture.