Scalar curvature rigidity of degenerate warped product spaces

TL;DR

This paper establishes scalar curvature extremality and rigidity results for a broad class of warped product spaces, including degenerate cases, answering longstanding questions posed by Gromov across all dimensions.

Contribution

It proves scalar curvature extremality and rigidity for degenerate warped product spaces with various leaves, extending Gromov's questions to all dimensions.

Findings

Scalar curvature extremality for degenerate warped products

Rigidity results for spaces with nonnegative curvature operators

Extension of Gromov's questions to all dimensions

Abstract

In this paper we prove the scalar curvature extremality and rigidity for a class of warped product spaces that are possibly degenerate at the two ends. The leaves of these warped product spaces can be any closed Riemannian manifolds with nonnegative curvature operators and nonvanishing Euler characteristics, flat tori, round spheres and their direct products. In particular, we obtain the scalar curvature extremality and rigidity for certain degenerate toric bands and also for round spheres with two antipodal points removed. This answers positively the corresponding questions of Gromov in all dimensions.