Quantification of scalar curvature under $C^0$ convergence using smoothing

TL;DR

This paper proves that a refined scalar curvature bound under $C^0$ convergence, conjectured by Gromov and verified in dimension three, holds in all dimensions three and above.

Contribution

It extends the refined scalar curvature bound under $C^0$ convergence to all dimensions greater than or equal to three.

Findings

Refined scalar curvature bound holds in all dimensions ≥ 3.

Constructed examples show the necessity of the refinement.

Confirmed Gromov's conjecture in higher dimensions.

Abstract

A quantitative version of the scalar lower bound under $C^0$ convergence was conjectured by Gromov. More recently, Mazurowski and Yao proved that a refined form of Gromov's conjecture holds in dimension three. Furthermore, they constructed examples demonstrating that such a refinement is necessary. In this paper, we establish that the refined quantitative bound holds in all dimensions greater than or equal to three.