The rigidity of sharp spectral gap in nonnegatively curved spaces

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.