Singular Structure of Minimal Surfaces

最小表面是肥皂膜的数学模型，或者更一般地，能量和面积成比例的任何界面：它们是局部最小化面积的表面。除了有着丰富的历史，极小曲面也是分析物理系统和几何问题的重要工具。一个突出的例子是稳定的极小曲面在广义相对论中标量曲率的研究中所起的作用。就像他们模拟的肥皂膜一样，极小的表面通常有一个不光滑的“奇点集”(例如，考虑浴缸中的气泡：它们有多个气泡相遇的奇点)。这个项目的目标是更好地理解极小曲面的奇异集，它具有什么结构，以及极小曲面在奇点附近的行为。除了从事这些研究项目外，PI还将指导研究生和本科生，并继续与当地社区中心针对小学、初中和高中学生的课外数学项目进行推广工作。这个项目研究面积最小、稳定或具有有限指数的高维极小超曲面。如果一个人排除了“Y-型”奇点(通常是由于可定向的自然原因)(在浴缸中遇到)，那么奇点集就降到余维7。该项目将研究具有孤立奇点和有限指数/面积的7维极小超曲面空间，以期证明某些奇异极小超曲面的“凸性”概念。PI还将调查非孤立奇点。这项研究还将尝试将欧氏空间中渐近于某些柱面锥的8维极小超曲面分类，这将用于模拟曲面向这些柱面奇点的几何退化。这项工作旨在根据最小超曲面的边界数据建立奇异集的一般等周型不等式。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Minimal surfaces are mathematical models of soap films, or more generally any interface whose energy and area are proportional: they are surfaces that locally minimize area. In addition to having a rich history, minimal surfaces are an important tool in analysis of physical systems and geometric questions. A prominent example is the role stable minimal surfaces play in the study of scalar curvature in general relativity. Just like the soap films they model, minimal surfaces in general have a "singular set" where they are not smooth (for example, consider bubbles in the bathtub: they have singular junctions were multiple bubbles meet). The goal of this project is to work towards better understanding of the singular set of a minimal surface, what structure it possesses, and the behavior of the minimal surface near singularities. In addition to pursuing these research projects, the PI will mentor graduate and undergraduate students and continue outreach work with a local community center's afterschool math program for elementary-, middle-, and high-school students.This project studies higher dimensional minimal hypersurfaces that are area-minimizing, stable, or have finite index. If one precludes, often for natural reasons of orientability, "Y-type" singularities (encountered in the bathtub), then the singular set drops to codimension 7. The project will study the space of 7-dimensional minimal hypersurfaces with isolated singularities and finite index/area in a closed 8-manifold, with the view towards a proving a notion of "bumpiness" for certain singular minimal hypersurfaces. The PI will also investigate non-isolated singularities. The research will also attempt to classify 8-dimensional minimal hypersurfaces in Euclidean space asymptotic to certain cylindrical cones, which would serve to model the geometric degeneration of surfaces towards these cylindrical singularities. The work aims to establish a general isoperimetric-type inequality for the singular set in terms of the minimal hypersurface's boundary data.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.