Stability for a class of three-tori with small negative scalar curvature

TL;DR

This paper introduces a class of Riemannian metrics on the three-torus and demonstrates that sequences with diminishing negative scalar curvature parts converge to flat metrics, highlighting stability properties.

Contribution

It establishes a stability result for a specific class of three-torus metrics with small negative scalar curvature using harmonic form inequalities.

Findings

Sequences with negative scalar curvature tending to zero converge to flat metrics.

The class of metrics is flexible and defined using harmonic one-forms.

Convergence is in the sense of Dong-Song.

Abstract

We define a flexible class of Riemmanian metrics on the three-torus. Then, using Stern's inequality relating scalar curvature to harmonic one-forms, we show that any sequence of metrics in this family whose negative part of the scalar curvature tends to zero in $L^2$ norm has a subsequence which converges to some flat metric on the three-torus in the sense of Dong-Song.