Ilaria Mondello

Synthetic Ricci lower bounds: new geometric examples

Singular manifolds appear naturally in geometry when considering quotients of smooth manifolds, their Gromov-Hausdorff limits or geometric flows. An important question in the study of such singular manifolds is to define a relevant notion of curvature, or curvature bounds. The work of Lott-Sturm-Villani and Ambrosio-Gigli-Savaré showed that it is possible to define a curvature-dimension condition on metric measure spaces, that corresponds to a Ricci lower bound in the case of smooth Riemannian manifolds. If some constructions on manifolds (quotients, cones, spherical suspension…) give examples of metric spaces satisfying the curvature-dimension condition, there is not any easy criterion to establish whether the RCD(K,N) condition holds on a manifold with simple singularities. In this talk, we present a geometric criterion for a compact stratified space to satisfy the RCD(K,N) condition: this gives a new large class of examples, including among others manifolds with conical singularities, isolated or not. The talk is based on a joint work with J. Bertrand, C. Ketterer and T. Richard.