Warped Tori with Almost Non-Negative Scalar Curvature

TL;DR

This paper proves that sequences of warped 3-torus metrics with scalar curvature approaching non-negativity, bounded volume and diameter, and a minimal surface area bound, converge to a flat 3-torus in geometric senses.

Contribution

It establishes convergence of warped 3-torus metrics with almost non-negative scalar curvature under specific bounds to a flat torus, extending understanding of geometric limits.

Findings

Sequences converge in Gromov-Hausdorff and Intrinsic Flat senses.

Limit space is a flat 3-torus.

Conditions ensure stability of geometric structure.

Abstract

For sequences of warped product metrics on a $3$-torus satisfying the scalar curvature bound $R_j \geq -\frac{1}{j}$, uniform upper volume and diameter bounds, and a uniform lower area bound on the smallest minimal surface, we find a subsequence which converges in both the Gromov-Hausdorff and the Sormani-Wenger Intrinsic Flat sense to a flat $3$-torus.