Synthetic Curvature Bounds via Optimal Transport Techniques

• Applicant: Dr. Christian Ketterer
• Subject Area: Mathematics
• Term: from 2017 to 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 396662902

Project A deals with a generalization of the concept of synthetic lower Ricci curvature bounds in the sense of Lott, Sturm and Villani. The aim is to introduce and study a new curvature-dimension condition that captures variable lower curvature bounds where the curvature function belongs to some Lp -space w.r.t. the reference measure. In particular, one of the project’s major goals is to prove compactness and stability properties of families of metric measure spaces in the topology of measured Gromov-Hausdorff convergence, and w.r.t. associated Lp weak convergence of the curvature functions. As consequences one will be able to establish almost rigidity and regularity statements that will be essentially new also for families of smooth spaces.Project B continues the analysis of new curvature conditions in the sense of optimal transport introduced by Ketterer and Mondino. These new conditions characterize p-intermediate Ricci, lower sectional curvature (and in some cases also upper sectional curvature) bounds for Riemannian manifolds via convexity of the relative entropy w.r.t. the Hp -Hausdorff measure. p-intermediate Ricci curvature is the trace of the Riemannian curvature tensor w.r.t. p-dimensional subspaces in the tangent space. Here, p is a natural number between zero and the space dimension. The statement only relies on the underlying metric structure. Hence, a very natural question that emerges from this characterization is if there is a connection to synthetic sectional curvature bounds in the sense of Alexandrov. Though a general equivalence to Alexandrov’s curvature condition cannot be expected, the project’s plan is to establish the result of Ketterer and Mondino in this context.Project C is about recently introduced Super-Ricci flow and Ricci flow conditions for metric measure space times. Still one of most important and most challenging problems in geometric analysis is to introduce a weak notion of Ricci flow for non-smooth initial data. First steps in this direction were taken by recent preprints of Sturm and Haslhofer-Naber. Sturm gives a time dependent generalization of his entropy convexity condition, Haslhofer and Naber apply a Bakry-Emery-type calculus on the path space. The project aims to draw connections between these concepts and previous results on optimal transport w.r.t. Perelman’s L-length and the Hamilton Harnack inequality.

• DFG Programme: Research Fellowships
• International Connection: Canada
• Hosts: Professor Dr. Robert Haslhofer; Professor Dr. Vitali Kapovitch