Ricci Flow

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

A Ricci flow is a geometric process that can be used to improve a given geometry towards a more homogeneous one. Ricci flows have gained increasing interest, as they have been used to prove various longstanding conjectures, such as the Poincaré and Geometrization Conjectures, as well as the Generalized Smale Conjecture in dimension 3. The general expectation is that a Ricci flow produces a geometry in the limit that is in some sense inherent to the topology, that is, the loose makeup, of the underlying space. However, usually a Ricci flow incurs complicated singularities in finite time. In dimension 3, these singularities can be removed manually by a so-called surgery construction and the flow can be continued beyond them. The long-term goal of this project is to generalize this surgery construction to dimension 4, and possibly higher. To achieve this, the PI will study the singularity formation of higher dimensional Ricci flows, using a new theory he recently found. A successful construction in dimension 4 may have interesting topological and geometric applications. The PI will also further study Ricci flows with surgery, and the closely related "Ricci flows through singularities," in dimension 3 and find further geometric and topological applications. The award provides funds for graduate students to engage in research related to the project. The research project is split into two parts. The first project is a continuation of the PI's recently obtained compactness and partial regularity theory for Ricci flows in higher dimensions. The goal of the project is to use this new theory to construct a "Ricci flow with surgery'" or "Ricci flow through singularities" in dimension 4, generalizing the analogous construction in dimension 3. The strategy for achieving this is to deduce spatial asymptotic estimates on blow-up limits and use these to obtain a qualitative picture of the singularity formation. Based on this picture, the next step is to remove singularities via cylindrical and conical surgery constructions. The project also aims to characterize the long-time asymptotics of the flow. A successful construction and analysis may lead to several interesting topological and geometric applications. The second project continues work of the PI and collaborator on the uniqueness and continuous dependence of 3-dimensional singular Ricci flows through singularities. Previous work used this continuous dependence to resolve the Generalized Smale Conjecture (which classifies diffeomorphism groups of certain 3-manifolds up to homotopy) and a conjecture regarding the space of metrics with positive scalar curvature on 3-manifolds. This project will study more deeply the techniques used in these proofs, which have the potential to produce further results. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.