Mean Curvature Flow and Singular Minimal Surfaces

这个项目涉及几何变分学，也就是说，研究在某种意义上对几何函数最优的对象的性质。变分问题出现在纯数学和应用数学的几个领域以及许多物理科学中。例如，最小表面作为区域功能的关键点出现，并提供肥皂膜的数学模型。另一个例子是子流形的熵函数，它可以作为表面复杂性的自然度量。用于研究该泛函的主要工具是平均曲率流，这是一个动态过程，粗略地说，它以一种尽可能快地减小其面积的方式不断地使表面变形。平均曲率流最初是作为材料科学中某些现象的模型来研究的，并且在计算机图形学和图像识别中也有应用。本项目旨在通过研究最小曲面，特别是那些奇异曲面，以及它们与子流形熵泛函的联系，来促进对这一领域的理解。该项目包括研究生的研究培训机会。本课题将研究奇异最小曲面和奇异平均曲率流的性质。一个特别的重点是探索，通过平均曲率流，如何冷- minicozzi熵与最小子流形的性质在各种设置。另一个探索的途径是奇异最小表面的研究，其奇异性是模仿在肥皂膜中观察到的。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.