Abstract. Obata's theorem characterizes the equality case in the spectral gap inequality for N dimensional Riemannian manifolds with Ricci curvature bounded below by N-1. In this talk I will present an approach to the quantitative study of the shape of eigenfunctions when equality is almost attained. The key tool is the so-called localization technique, which allows to treat possibly non smooth spaces.
The talk is based on a joint work with Fabio Cavalletti (SISSA) and Andrea Mondino (University of Oxford).