A theorem of Lichnerowicz (1958) states that the spectral gap (or sharp Poincaré constant) of a smooth n-dimensional Riemannian manifold with curvature bounded from below by $n-1$ is bounded by $n$, which is the spectral gap of the unit $n$-sphere. This bound has since been extended to metric-measure spaces satisfying a curvature-dimension condition. In this talk, I will present a result on stability of the bound: if a space has almost minimal spectral gap, then the pushforward of the volume measure by a normalized eigenfunction is close to a Beta distribution with parameter $n/2$, with a sharp estimate on the $L^1$ optimal transport distance. Joint work with Ivan Gentil and
Jordan Serres
Paul Pegon
