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Ricci flows for non-smooth spaces, monotonic quantities, and rigidity

One of the main objectives of this project is to develop a synthetic theory of Ricci flows for non-smooth spaces in the general framework of time evolutions of metric measures spaces. This setting will allow for a broad range of applications including in particular previously studied scenarios like smooth Ricci flows starting from singular initial data or flows on manifolds with isolated singularities. We will use the heat flow and ideas from optimal transport to develop several possible synthetic notions of Ricci flows building on the well developed theory of mm spaces with synthetic curvature bounds, our existing approaches to super Ricci flows, as well as our recent results on upper control on the Ricci curvature in the static case. In the metric measure context, the evolution of the measure presents an additional degree of freedom. We will therefore consider N-Ricci flows for a dimension parameter N which in the case of a smooth family of weighted Riemannian manifolds are equivalent to the weighted Ricci flow equation. Of particular interest will be n-Ricci flows, where n is taken to be the actual dimension of the space, in which case the measure is forced to evolve essentially as the Hausdorff measure of the metric. A second major goal of this project will be to exhibit functionals that behave monotonically under Ricci flow for metric measure spaces such as analogues of Perelman's W-entropy and L-distance. Once monotonicity is established, an important question will be about rigidity i.e. the characterization of spaces where strict monotonicity fails. Another major objective of this project will be to build to establish sharp geometric and analytic estimates for time-dependent metric measure spaces such as parabolic versions of the Laplace comparison theorem for the L distance or differential Harnack inequalities for solutions to the heat equation.


Publications


    Team Members

    Prof. Dr. Matthias Erbar
    Project leader
    Universität Bielefeld

    Karl-Theodor Sturm
    Project leader

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