Scalar curvature rigidity of warped product metrics

TL;DR

This paper proves new scalar curvature rigidity results for warped product metrics and spheres with points removed, extending previous work and solving a longstanding problem using spin geometry techniques.

Contribution

It generalizes scalar curvature rigidity results to all dimensions for warped products and spheres with antipodal points removed, resolving a problem posed by Gromov.

Findings

Rigidity of warped products with log-concave warping functions

Scalar curvature rigidity of spheres with antipodal points removed

Extension of previous results to all dimensions

Abstract

We show scalar-mean curvature rigidity of warped products of round spheres of dimension at least 2 over compact intervals equipped with strictly log-concave warping functions. This generalizes earlier results of Cecchini-Zeidler to all dimensions. Moreover, we show scalar curvature rigidity of round spheres of dimension at least 3 with two antipodal points removed. This resolves a problem in Gromov's ''Four Lectures'' in all dimensions. Our arguments are based on spin geometry.