Gap phenomena under curvature restrictions

TL;DR

This paper explores various gap phenomena related to Ricci and sectional curvature, including spectral gaps, metric-rigidity, and topological-rigidity, and proposes open problems in these areas.

Contribution

It introduces a unified discussion of three types of gap phenomena under curvature restrictions and suggests open problems for future research.

Findings

Analysis of spectral gaps under curvature constraints

Discussion of sharp gap metric-rigidity results

Proposal of open problems in topological-rigidity

Abstract

In the paper we discuss gap phenomena of three different types related to Ricci (and sectional) curvature. The first type is about spectral gaps. The second type is about sharp gap metric-rigidity, originally due to Anderson. The third is about sharp gap topological-rigidity. We also propose open problems along these directions.