Smoothing and non-smoothing via Ricci Flow

• Applicant: Dr. Franziska Borer, since 10/2018
• Subject Area: Mathematics
• Term: from 2018 to 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 408275461

It is well-known that for the Ricci flow with smooth initial data, finite time singularity formation is somehow a generic property. In the absence of a weak formulation, continuing the Ricci flow past a singular time has always been done by surgery, which depends on a number of non-arbitrary choices. In order to avoid them and define a canonical Ricci flow through singularities, an important first step (which is the detonating question of this proposal) is to develop a theory to produce solutions of the Ricci flow starting at a possibly singular metric space. Our proposal includes a short time existence result for the Ricci flow with much weaker assumptions on the initial condition than those required in the previous literature and a detailed strategy for applications to smoothen out a variety of singular spaces. As a by-product we design a detailed strategy to solve an open problem (Smoothing Conjecture) in metric geometry.In the particular case of two dimensions, much stronger results can be expected if we exploit the analogy with the harmonic map flow.Concretely we propose to improve the current well-posedness theory starting with rough initial data. We have a double objective: byimposing volume-preservation we plan to show that the solution becomes instantaneously smooth but by violating such a preservation we suggest how to construct solutions where non-smoothing happens. The latter would mean the first example of a non-smooth (and non-unique) solution of Ricci flow starting from a closed surface.

• DFG Programme: Research Grants
• Ehemalige Antragstellerin: Professorin Dr. Esther Cabezas-Rivas, until 9/2018