The rigidity of sharp spectral gap in non-negatively curved spaces

We extend the celebrated rigidity of the sharp first spectral gap under Ric >= 0 to compact infinitesimally Hilbertian spaces with non-negative (weak, also called syn-thetic) 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 1 = pi 2/diam2 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.(c) 2022 Elsevier Ltd. All rights reserved.