#	Pagina
attuale pagina	/open-h2020/projects/204827/results.html

Report

Teaser, summary, work performed and final results

Periodic Reporting for period 2 - RicciBounds (Metric measure spaces and Ricci curvature â€” analytic, geometric, and probabilistic challenges)

Teaser

Summary

The project is devoted to innovative directions of research on metric measure spaces (`mm-spaces\') and synthetic bounds for the Ricci curvature.

(i) One of the project\'s main goals is to bring together two - previously unrelated - areas of mathematics which both have seen an impressive development in the last decade: the study of `static\' mm-spaces with synthetic Ricci bounds and the study of Ricci flows for `smooth\' Riemannian manifolds.

(ii) Another major aim is to push forward the analytic and geometric calculus on metric measure spaces beyond the scope of spaces with uniform lower bounds for the Ricci curvature towards spaces with measure-valued curvature bounds.

(iii) Furthermore, the project aims to initiate the development of stochastic calculus on mm-spaces and, in particular, to provide pathwise insights into the effect of (singular) Ricci curvature.

Work performed

(i) The novel concept of dynamical convexity allowed us to to merge two cutting-edge developments (the study of `static\' mm-spaces with synthetic Ricci bounds and the study of Ricci flows for `smooth\' Riemannian manifolds) and to develop a theory of super Ricci flows for metric measure spaces. Moreover, we analyzed in great detail the heat flow on time-dependent metric measure spaces and characterized super-Ricci flows in terms of contraction properties of heat flows and Brownian motions on such time-dependent spaces.

(ii) A first and important step is the detailed analysis of mm-spaces with variable lower bounds for the Ricci curvature. Here major progress has been made, including the important proof of the equivalence of the Eulerian and of the Lagrangian approach to such synthetic curvature bounds. Also first breakthroughs have been achieved concerning measure-valued curvature bounds arising from the curvature of the boundary in the analysis of the Neumann heat flow. Transformation formulas for Ricci bounds under conformal transformations and under time changes turned out to be of great importance.

(iii) Probabilistic representations turned out to provide deep insights into the effects of variable curvature for gradient estimates and for transport estimates. This in particular also applies to the Neumann heat flow, taking into account also the effects of the curvature of the boundary. One of the major achievements is the construction of optimally coupled pairs of Brownian motions.

Final results

(i) In the time-independent (\'static\') case, there is a whole zoo of functional inequalities which are equivalent to (uniform) lower bounds for the Ricci curvature. We will formulate such functional inequalities also in the time-dependent case and prove their equivalences.
We also aim to combine the concept of super-Ricci flows with our concept of synthetic upper Rice bounds to derive new insights for \'Ricci flows of mm-spaces\'.
Moreover, we plan to study in more details the heat flow on general time-dependent mm-spaces without any regularity assumptions on the time dependence of distances and measures.

(ii) We will push forward the analysis of mm-spaces with measure-valued curvature bounds - both in the Eulerian and in the Lagrangian setting. If possible, we will prove the equivalence of both approaches. Particular emphasis will be given to singular \'Ricci\' curvature arising from the curvature of boundaries and from gluing effects.

(iii) The stochastic calculus will be pushed forward towards a probabilistic representations of the heat semigroup acting on 1-forms.