Stability of the positive mass theorem under Ricci curvature lower bounds

TL;DR

This paper proves that sequences of asymptotically flat 3-manifolds with nonnegative scalar curvature and Ricci curvature bounded below, whose ADM mass approaches zero, converge to Euclidean space, confirming a stability conjecture.

Contribution

It establishes Gromov-Hausdorff stability of the positive mass theorem under Ricci curvature lower bounds, confirming a conjecture by Huisken and Ilmanen.

Findings

Sequences with ADM mass approaching zero converge to Euclidean space

Stability holds under Ricci and scalar curvature bounds

Uses harmonic level set and almost splitting techniques

Abstract

We establish Gromov-Hausdorff stability of the Riemannian positive mass theorem under the assumption of a Ricci curvature lower bound. More precisely, consider a class of orientable complete uniformly asymptotically flat Riemannian 3-manifolds with nonnegative scalar curvature, vanishing second homology, and a uniform lower bound on Ricci curvature. We prove that if a sequence of such manifolds has ADM mass approaching zero, then it must converge to Euclidean 3-space in the pointed Gromov-Hausdorff sense. In particular, this confirms Huisken and Ilmanen's conjecture on stability of the positive mass theorem under the assumptions described above. The proof is based on the harmonic level set approach to proving the positive mass theorem, combined with techniques used in the proof of Cheeger and Colding's almost splitting theorem. Furthermore, we show that the same results hold under a more general lower bound on scalar curvature.