Ancient Solutions to Geometric Flows

Geometric flows are evolution equations that describe motions of surfaces, or higher dimensional analogues, with speeds determined by their curvatures. A flow of principal concern for this project is the Mean Curvature Flow, characterized by the property that it evolves a surface so that its area decreases as rapidly as possible. Geometric flows have extensive applications to various physical problems.

For instance, mean curvature flow occurs in the description of the evolution of interfaces arising in several multiphase physical models. Moreover, it is used in material science to model cell, grain and bubble growth. Geometric flows have also proven to have extremely useful geometric applications, specifically classification theorems and geometric inequalities.

This project will focus on ancient solutions to mean curvature flow, these being solutions that have existed for all times in the past. A large and important class of these solutions has hitherto resisted attempts at elucidation, but a new method of constructing, studying and classifying them will be brought to bear. The project also includes training of graduate students as well as the organization of workshops and summer schools to help create opportunities for students to be exposed to new developments in geometric analysis.

Ancient solutions have an important role in the study of singularities; these constitute an obstruction to the existence for all times of the flow, and it is therefore of major interest to understand their geometry and behavior. As is typically the case in mathematics, focus has to be narrowed to some extent in order to obtain satisfying answers. Therefore this project will focus on ancient solutions to mean curvature flows that are confined to slab regions.

This project will construct, in all dimensions, a large family of new examples, both symmetric and asymmetric; the project will also construct many eternal examples which are not of the standard kind that evolve by translation. Additionally, the project proposes a novel way of classifying such solutions. Finally, the project plans to apply results to the non-collapsed setting to obtain properties of entire solutions.

Looking further ahead, there is strong evidence that these methods apply to a much wider class of geometric flows. The project includes plans to run seminars, train and mentor students and organize workshops on the subject of flows. The PI also aims to broaden participation of under-represented groups through various activities.

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.