Synthetic Ricci Curvature Bounds and the Regularity of Optimal Transport Maps on Metric Measure Spaces

This monograph investigates the existence, uniqueness, and regularity properties of optimal transport maps within the framework of metric measure spaces satisfying synthetic Ricci curvature bounds, specifically the RCD*(K,N) condition. We analyze the structural implications of the curvature-dimension condition introduced by Lott, Sturm, and Villani, refined by the Riemannian property established by Ambrosio, Gigli, and Savaré. Central to this work is the study of the Monge problem in non-smooth settings, where the classical regularity theory of Caffarelli cannot be directly applied. We provide a rigorous exposition of the existence of optimal maps in essentially non-branching spaces and discuss the Sobolev regularity of these maps. Furthermore, we explore the connection between the semi-concavity of Kantorovich potentials and the geometric rigidity of the underlying space, demonstrating how lower curvature bounds enforce a weak form of regularity on the transport plan, thereby extending the classical theory of optimal transportation to the non-smooth geometric analytic setting.