Stability of the positive mass theorem and torus rigidity theorems under integral curvature bounds

TL;DR

This paper investigates the stability of the positive mass theorem and torus rigidity under integral curvature bounds by analyzing harmonic maps and establishing quantitative closeness to model spaces.

Contribution

It introduces new stability results for the positive mass theorem and Geroch conjecture using integral Ricci curvature and isoperimetric bounds.

Findings

Small mass implies diffeomorphism to flat space

Almost non-negative scalar curvature leads to H"older closeness

Quantitative control of harmonic maps under curvature bounds

Abstract

Work of D. Stern and Bray-Kazaras-Khuri-Stern provide differential-geometric identities which relate the scalar curvature of Riemannian 3-manifolds to global invariants in terms of harmonic functions. These quantitative formulas are useful for stability results and show promise for more applications of this type. In this paper, we analyze harmonic maps to flat model spaces in order to address conjectures concerning the geometric stability of the positive mass theorem and the Geroch conjecture. By imposing integral Ricci curvature and isoperimetric bounds, we leverage the previously mentioned formulas to establish strong control on these harmonic maps. When the mass of an asymptotically flat manifold is sufficiently small or when a Riemannian torus has almost non-negative scalar curvature, we upgrade the maps to diffeomorphisms and give quantitative H\"older closeness to the model spaces.