Ancient Solutions and Singularities in Geometric Flows

The focus of the project is to study singularity formation in various geometric flows. These flows are characterized by the deformation of geometric objects such as metrics, mappings, and submanifolds by geometric quantities such as curvature and consist of partial differential equations of parabolic type. Geometric flows appear in many real world applications.

For example, surface tension along moving interfaces in fluids and materials is proportional to mean curvature; mean curvature flow and affine mean curvature flow are useful for image processing. However, studying geometric equations can be challenging due to nonlinearities and the possible development of singularities, especially topological changes.

One way to understand those singularities is to zoom in and understand how solutions look as they approach the singular time after which a smooth solution no longer exists. During this limiting process we get special solutions to a geometric equation that are called ancient solutions, which have existed for an infinite amount of time in the past. Understanding those solutions could be useful in obtaining more topological and geometric information about a geometric object. The project will also include training of students and the mentoring of junior researchers.

The aim of the project is the classification of ancient solutions to nonlinear geometric flows, such as, the Ricci flow and the mean curvature flow. This project will combine the PDE techniques and geometric estimates to study ancient solutions of these flows. The goal is to classify ancient closed noncollapsed solutions to higher dimensional Ricci flow (cases n = 2, 3 have been solved), under the assumption that solution becomes asymptotically cylindrical as time approaches minus infinity.

One motivation for this classification comes from showing an analogue of the Mean Convex Neighborhood Theorem for the Ricci flow. This could potentially enable us to perform surgery in the Ricci flow in higher dimensions without assuming global curvature conditions initially. As a continuation of a completed project with collaborators, PI will investigate noncollapsed ancient solutions to the mean curvature flow that are asymptotic to other generalized round cylinders besides the well understood case of a round cylinder.

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.