A spectral lower bound in singular setting

 A very well known result in Riemannian geometry, the Obata-Lichnerowicz theorem, relates the Ricci curvature and the spectrum of the Laplacian: for a compact Riemannian manifold of dimension n, if the Ricci tensor is bounded below by $(n-1)$, then the first non-zero eigenvalue of the Laplacian is larger than or equal to the dimension of the manifold. Equality holds if and only if the manifold is isometric to the sphere. We will show how an analogue result holds in the singular setting of stratified spaces, which are metric spaces generalizing the notion of isolated conical singularities. The last part of the talk is devoted to the consequences of this result on the existence of a constant scalar curvature metric.