Modul: MAT971 Stochastische Prozesse

## Stability of the spectral gap under a curvature-dimension condition

Talk by Prof. Dr. Max Fathi

Speaker invited by: Dr. Gaultier Lambert

**Date:** 02.03.22 **Time:** 17.15 - 18.15 **Room:** Y27H12

A theorem of Lichnerowicz (1958) states that the spectral gap (or sharp Poincare 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 L1 optimal transport distance. Joint work with Ivan Gentil and Jordan Serres.