Resultate zur Rigidität in CD(K,N) Räumen mit negativem N

Wider research context: An important class of singular spaces is the one of metric measure spaces with generalized lower bounds on the Ricci curvature formulated in terms of optimal transport. The curvature-dimension condition CD(K,N) depends on two parameters K and N, playing the role of a curvature and a dimension bound, respectively. The class of spaces satisfying the CD(K,N) condition with N > 0 has been a central object of investigation in the last years. In particular, the validity of several analytic and geometric properties was proven in the setting of CD(K,N) spaces. Objectives: This project is focused on the study of the geometry of CD spaces with negative dimension, extending the results already established for weighted Riemannian manifolds satisfying this condition, through the use of novel techniques. In order to do so, I propose an Eulerian description of the CD(K, N ) condition with N < 0, based on the validity of a suitably-defined Bochner inequality, which is well-known to be a fundamental tool in the study of the geometry of the space. Approach: In the framework of CD(K,N) spaces with negative values of N, I aim at proving: (1) the possibility to define a suitable energy functional in this setting for which the reference measure presents some pathologies; (2) the possibility to provide a meaningful version of the Bochner inequality; (3) the validity of certain rigidity properties of these structures. Innovation: The class of CD(K,N) spaces with negative dimension covers wider/wilder structures than the ones with N > 0. Therefore I will obtain new results in the context of the analysis of CD spaces and provide new techniques to study geometry on very general metric measure spaces. Researchers involved: The program will be developed by an international research team coordinated by Chiara Rigoni with supervision of Professor Ulisse Stefanelli (Hosting Mentor) and the cooperation of Professor Maria Gordina (Storrs) and Dr. Kohei Suzuki (Bielefeld).