Ricci curvature bounds for warped products and cones

Abstract

In this thesis we prove generalized lower Ricci curvature bounds in the sense of optimal transport for warped products and cones over metric measure spaces, and we prove a maximal diameter theorem in this context.<br /> In the first part we focus on the case when the underlying spaces are complete Riemann-Finsler manifolds equipped with a smooth reference measure. The proof is based on calculations for the N-Ricci tensor and on the study of optimal transport of absolutely continuous probability measures in warped products. On the one hand, this result covers a theorem of Bacher and Sturm concerning Euclidean and spherical N-cones. On the other hand, it can be seen in analogy to a result of Bishop and Alexander in the setting of Alexandrov spaces with curvature bounded from below. Because the warped product metric can degenerate we regard a warped product as a singular metric measure space that is in general neither a Finsler manifold nor an Alexandrov space again but a space satisfying a curvature-dimension condition in the sense of Lott, Sturm and Villani.<br /> In the second part we treat the case of general metric measure spaces. The main result states that the Euclidean cone over any metric measure space satisfies the reduced Riemannian curvature-dimension condition RCD*(0,N+1) if and only if the underlying space satisfies RCD*(N-1,N). The proof uses a characterization of reduced Riemannian curvature-dimension bounds by Bochner’s inequality that was established for general metric measure spaces by Erbar, Kuwada and Sturm and announced independently by Ambrosio, Mondino and Savaré. By application of this result and the Gigli-Cheeger-Gromoll splitting theorem we prove a maximal diameter theorem for metric measure spaces that satisfy the reduced Riemannian curvature-dimension condition. This generalizes the classical maximal diameter theorem for Riemannian manifolds which was proven by Cheng.