Rigidity Results Involving Stabilized Scalar Curvature

TL;DR

This paper proves a rigidity theorem related to a recent systolic inequality involving stabilized scalar curvature, using foliations and Ricci flow techniques to extend classical results to this new setting.

Contribution

It introduces a novel approach combining foliations and Ricci flow to establish scalar curvature rigidity in the context of stabilized scalar curvature.

Findings

Established a new rigidity theorem for stabilized scalar curvature.

Generalized classical scalar curvature rigidity results to the stabilized setting.

Developed a monotone quantity using Ricci flow and heat equation techniques.

Abstract

We establish a rigidity theorem for Brendle and Hung's recent systolic inequality, which involves Gromov's notion of \(T^{\rtimes}\)-stabilized scalar curvature. Our primary technique is the construction of foliations by free boundary weighted constant mean curvature hypersurfaces, enabling us to generalize several classical scalar curvature rigidity results to the \(T^{\rtimes}\)-stabilized setting. Additionally, we develop a monotone quantity using Ricci flow coupled with a heat equation, which is essential for rigidity analysis.