Mean Curvature Flow and Singular Minimal Surfaces

平均曲率流和奇异极小曲面

• 批准号: 2203132
• 负责人: Jacob Bernstein
• 金额: $ 23.44万
• 依托单位: Johns Hopkins University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-08-01 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2203132&HistoricalAwards=false
• 关键词: Mean; Curvature; Flow; Singular; Minimal

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.

这个项目关注的是变分的几何演算，也就是说，在某种意义上对几何泛函最优的对象的性质的研究。变分问题出现在一些领域的纯数学和应用数学和许多物理科学。例如，最小表面作为面积泛函的临界点出现，并提供了肥皂膜的数学模型。另一个例子是子流形的熵泛函，它是曲面复杂性的自然度量。用于研究此功能的主要工具是平均曲率流，这是一个动态过程，粗略地说，不断变形的表面的方式，尽可能快地减少其面积。平均曲率流最初是作为材料科学中某些现象的模型进行研究的，并且在计算机图形学和图像识别中也有应用。该项目旨在通过研究极小曲面，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。该项目将研究奇异极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Colding-Minicozzi熵涉及到各种设置中的极小子流形的属性。 另一个探索途径是奇异极小曲面的研究，其奇异性是以肥皂膜中观察到的奇异性为模型的。该奖项反映了NSF的法定使命，并被认为值得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来支持。