Non-Compact Solutions to Geometric Flows

Massachusetts Institute Of Technology, Cambridge MA

Investigators

Abstract

Geometric flows are processes that evolve surfaces or higher-dimensional spaces by their curvatures. In particular, the flows considered in this project are differential equations that model how the shape of a geometric object changes as its area, volume, or some other geometric quantity decreases as rapidly as possible. For example, the surface area decreases most rapidly under the mean curvature flow (MCF), while the enclosed volume deceases most rapidly under the Gauss curvature flow (GCF). Due to these natural decreasing properties, as the flow becomes singular (i.e., the object develops folds, corners, or other points of high curvature) these evolutions often tend (under magnification) toward optimal shapes minimizing the corresponding energies such as area and volume. For example, a soap bubble is the shape of a rescaled singularity of the MCF. These energy minimizers appear not only in geometry, but also in economics and physics; for instance, optimal transport refers to a mapping from one area to another that minimizes an energy which is the total cost of resource allocation. Thus, studying singularity of the geometric flows sheds new insight on the?understanding of energy minimizers?in physics and economics. This project aims to understand the singularity of various geometric flows including the Gauss curvature flow (GCF), the mean curvature flow (MCF), and the Ricci flow (RF). For the MCF and the RF, the uniqueness of non-collapsed type II ancient solutions will be considered. For the GCF, the existence and the uniqueness of type II closed ancient solutions will be studied under suitable conditions.?Moreover, this project also examines the convergence to the translating solutions to the curve-shortening flow and the GCF.? Interior estimates, decay rate, and monotonicity formulas will be developed.?In addition, this project will also address optimal regularity and free boundary problems for the GCF and and other fully non-linear equations, including optimal transport and Monge-Ampere equations. This project will provide a method to utilize prescribed singularity conditions to obtain optimal regularity and free boundary regularity, in particular in free boundary problems that arise from quantitative economics and classical mechanics. 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.