Conformal tori with almost non-negative scalar curvature

TL;DR

This paper proves that a sequence of conformal metrics with almost non-negative scalar curvature on a torus converges to a flat metric in multiple geometric senses, under certain uniform control conditions.

Contribution

It establishes convergence of conformal metrics with almost non-negative scalar curvature to flat metrics in various geometric senses, given uniform geometric control.

Findings

Conformal metrics with almost non-negative scalar curvature converge to flat metrics.

Convergence occurs in volume preserving intrinsic flat, $L^{p}$, and measured Gromov-Hausdorff senses.

Results depend on uniform control of the geometry of the metrics.

Abstract

In this work, we consider sequence of metrics with almost non-negative scalar curvature on torus. We show that if the sequence is uniformly conformal to another sequence of metrics with uniformly controlled geometry, then it converges to a flat metric in the volume preserving intrinsic flat sense, $L^{p}$ sense and the measured Gromov-Hausdorff sense.