Almost non-negative scalar curvature on Riemannian manifolds conformal to tori

TL;DR

This paper advances the understanding of scalar curvature rigidity by proving cases where manifolds conformal to tori or negatively curved manifolds approximate flat tori, linking geometric stability to the Yamabe problem.

Contribution

It reduces the scalar torus rigidity conjecture to the conformal case and proves it for sequences conformal to flat or negatively curved tori converging to flat tori.

Findings

Proved the conjecture for conformal sequences to flat tori.

Extended results to negatively curved manifolds converging to flat tori.

Discussed the full conjecture from the conformal perspective.

Abstract

In this article we reduce the geometric stability conjecture for the scalar torus rigidity theorem to the conformal case via the Yamabe problem. Then we are able to prove the case where a sequence of Riemannian manifolds is conformal to a uniformly controlled sequence of flat tori and satisfies the geometric stability conjecture. We are also able to handle the case where a sequence of Riemannian manifolds is conformal to a sequence of constant negative scalar curvature Riemannian manifolds which converge to a flat torus in $C^1$. The full conjecture from the conformal perspective is also discussed as a possible approach to resolving the conjecture.