Mean Curvature Flow and Singular Minimal Surfaces

点击翻译按钮获取中文摘要

This project concerns the geometric calculus of variations, that is, the study of the properties of objects that are optimal in some sense for a geometric functional. Variational questions arise in several areas of pure and applied mathematics and in many physical sciences. For example, minimal surfaces arise as critical points of the area functional and provide a mathematical model of soap films. Another example is an entropy functional for submanifolds that serves as a natural measure of the complexity of a surface. The main tool used to study this functional is the mean curvature flow, which is a dynamic process that, roughly speaking, continuously deforms a surface in a manner that decreases its area as quickly as possible. Mean curvature flow was first studied as a model of certain phenomena in materials science and has also found applications in computer graphics and image recognition. This project aims to advance understanding in this area through study of minimal surfaces, especially those that are singular, and their connections to the submanifold entropy functional. The project includes opportunities for research training of graduate students.This project will study properties of singular minimal surfaces and singular mean curvature flows. A particular focus is on exploring, via the mean curvature flow, how Colding-Minicozzi entropy relates to properties of minimal submanifolds in a variety of settings. Another avenue of exploration is the study of singular minimal surfaces whose singularities are modeled on those observed in soap films.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.