The rigidity of sharp spectral gap in nonnegatively curved spaces

TL;DR

This paper proves a rigidity result for the first spectral gap in non-negatively curved metric measure spaces, showing it is achieved only in one-dimensional cases, and introduces new techniques involving Sobolev theory and localization.

Contribution

It extends spectral gap rigidity to synthetic Ricci curvature spaces, including various non-smooth geometries, using novel methods that could be broadly applicable.

Findings

Spectral gap equals rac{}{ ext{diam}^2} only in one-dimensional spaces with constant density

New techniques combining Sobolev theory and singular 1D localization

Almost rigidity results derived from the main rigidity theorem

Abstract

We extend the celebrated rigidity of the sharp first spectral gap under $Ric\ge0$ to compact infinitesimally Hilbertian spaces with non-negative (weak, also called synthetic) Ricci curvature and bounded (synthetic) dimension i.e. to so-called compact $RCD(0,N)$ spaces; this is a category of metric measure spaces which in particular includes (Ricci) non-negatively curved Riemannian manifolds, Alexandrov spaces, Ricci limit spaces, Bakry-\'Emery manifolds along with products, certain quotients and measured Gromov-Hausdorff limits of such spaces. In precise terms, we show in such spaces, $\lambda = \frac{\pi^{2}}{\mathrm{diam}^2}$ if and only if the space is one dimensional with a constant density function. We use new techniques mixing Sobolev theory and singular $1D$-localization which might also be of independent interest. As a consequence of the rigidity in the singular setting, we also derive almost rigidity results.