Periodic Reporting for period 4 - RicciBounds (Metric measure spaces and Ricci curvature — analytic, geometric, and probabilistic challenges)
Okres sprawozdawczy: 2021-03-01 do 2022-11-30
The project was 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 was to bring together two -- previously unrelated - areas of mathematics which both have seen an impressive development in the previous 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 was 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 aimed to initiate the development of stochastic calculus on mm-spaces and, in particular, to provide pathwise insights into the effect of (singular) Ricci curvature.
(i) One of the project's main goals was to bring together two -- previously unrelated - areas of mathematics which both have seen an impressive development in the previous 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 was 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 aimed to initiate the development of stochastic calculus on mm-spaces and, in particular, to provide pathwise insights into the effect of (singular) Ricci curvature.
(i) The novel concept of dynamical convexity allowed us 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 numerous functional inequalities on such time-dependent, singular spaces. Even in the smooth time-dependent case and also in the static case, several of these characterizations were unknown before.
The gradient flow perspective on the space of probability measures was successfully studied for the heat flow in various other cases, including quantum heat semigroup, heat flow with Neumann boundary conditions and – mostly surprisingly – also heat flow with Dirichlet boundary conditions.
(ii) With “tamed spaces”, a novel direction of research for metric measure spaces with distribution-valued Ricci bounds was established. This in particular covers the important case of measure-valued curvature bounds arising from the curvature of the boundary in the analysis of the Neumann heat flow on non-convex domains. To deal with the non-convexity, the innovative “convexification method” has been developed. Moreover, novel transformation formulas for the parameters K and N in the curvature-dimension condition CD(K,N) under conformal transformations and under time changes of the underlying metric measure spaces turned out to be of great importance.
Furthermore, first and second order calculus on metric measure spaces could be developed further on spaces with synthetic lower Ricci bounds, and various novel concepts of synthetic upper Ricci bounds were presented and analyzed in detail. As important applications, new rigidity results could be obtained.
(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 (“singular”) curvature of the boundary. Of particular interest is the construction of optimally coupled pairs of Brownian motions. Furthermore, fundamental results concerning the equivalence of Eulerian and Lagrangian formulations of the curvature-dimension condition on metric measure spaces with variable Ricci bounds have been obtained. A more detailed analysis based on a pathwise calculus was presented on manifolds with Kato-class Ricci bounds.
Related results concern the short time asymptotic of the heat content, again depending in a sophisticated way on the curvature of the boundary.
Numerous further important results at the crossroad of probabilistic objects and curvature concepts were obtained, among them limit theorems on Poisson spaces, spectral estimates on random Riemannian manifolds, and conformally invariant Liouville quantum geometries.
The gradient flow perspective on the space of probability measures was successfully studied for the heat flow in various other cases, including quantum heat semigroup, heat flow with Neumann boundary conditions and – mostly surprisingly – also heat flow with Dirichlet boundary conditions.
(ii) With “tamed spaces”, a novel direction of research for metric measure spaces with distribution-valued Ricci bounds was established. This in particular covers the important case of measure-valued curvature bounds arising from the curvature of the boundary in the analysis of the Neumann heat flow on non-convex domains. To deal with the non-convexity, the innovative “convexification method” has been developed. Moreover, novel transformation formulas for the parameters K and N in the curvature-dimension condition CD(K,N) under conformal transformations and under time changes of the underlying metric measure spaces turned out to be of great importance.
Furthermore, first and second order calculus on metric measure spaces could be developed further on spaces with synthetic lower Ricci bounds, and various novel concepts of synthetic upper Ricci bounds were presented and analyzed in detail. As important applications, new rigidity results could be obtained.
(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 (“singular”) curvature of the boundary. Of particular interest is the construction of optimally coupled pairs of Brownian motions. Furthermore, fundamental results concerning the equivalence of Eulerian and Lagrangian formulations of the curvature-dimension condition on metric measure spaces with variable Ricci bounds have been obtained. A more detailed analysis based on a pathwise calculus was presented on manifolds with Kato-class Ricci bounds.
Related results concern the short time asymptotic of the heat content, again depending in a sophisticated way on the curvature of the boundary.
Numerous further important results at the crossroad of probabilistic objects and curvature concepts were obtained, among them limit theorems on Poisson spaces, spectral estimates on random Riemannian manifolds, and conformally invariant Liouville quantum geometries.
Most important breakthroughs including novel concepts, deep insights and fundamental results -- far beyond the previous state of the art and in some cases completely unexpected -- have been obtained within the project for
(1) Super-Ricci flows of metric measure spaces
(2) Distributional valued lower Ricci bounds
(3) Synthetic upper Ricci bounds
(4) Transformation formulas for the curvature-dimension parameters
(5) Optimal transport approach to heat flow with Dirichlet boundary conditions and gluing of metric measure spaces.
(1) Super-Ricci flows of metric measure spaces
(2) Distributional valued lower Ricci bounds
(3) Synthetic upper Ricci bounds
(4) Transformation formulas for the curvature-dimension parameters
(5) Optimal transport approach to heat flow with Dirichlet boundary conditions and gluing of metric measure spaces.