Metric measure spaces and synthetic Ricci bounds: Fundamental concepts and recent developments

Abstract

Metric measure spaces with synthetic Ricci bounds have attracted great interest in recent years, accompanied by spectacular breakthroughs and deep new insights. In this survey, I will provide a brief introduction to the concept of lower Ricci bounds as introduced by Lott–Villani and myself, and illustrate some of its geometric, analytic, and probabilistic consequences, among them Li–Yau estimates, coupling properties for Brownian motions, sharp functional and isoperimetric inequalities, rigidity results, and structural properties like rectifiability and rectifiability of the boundary. In particular, I will explain its crucial interplay with the heat flow and its link to the curvature-dimension condition formulated in functional-analytic terms by Bakry–Émery. This equivalence between the Lagrangian and the Eulerian approach then will be further explored in various recent research directions: (i) time-dependent Ricci bounds which provide a link to (super-) Ricci flows for singular spaces, (ii) second-order calculus, upper Ricci bounds, and transformation formulas, (iii) distribution-valued Ricci bounds which, e.g., allow singular effects of non-convex boundaries to be taken into account.