Singular Structure of Minimal Surfaces

最小表面是肥皂膜的数学模型，或者更一般地说，是能量和面积成正比的任何界面：它们是局部面积最小的表面。除了具有丰富的历史外，最小曲面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极小曲面在广义相对论中研究标量曲率中所起的作用。就像他们模拟的肥皂膜一样，最小表面通常有一个“奇异集”，在那里它们不光滑（例如，考虑浴缸里的气泡：它们有多个气泡相遇的奇异连接点）。这个项目的目标是为了更好地理解最小曲面的奇异集，它拥有什么结构，以及最小曲面在奇点附近的行为。除了进行这些研究项目外，PI还将指导研究生和本科生，并继续与当地社区中心为小学、初中和高中学生开设的课后数学课程开展拓展工作。本项目研究高维最小超曲面，这些曲面是面积最小化的、稳定的或具有有限索引的。如果排除了“y型”奇点（通常是由于可定向性的自然原因），那么奇异集就降为余维数7。该项目将研究封闭8流形中具有孤立奇点和有限指数/面积的7维极小超曲面空间，旨在证明某些奇异极小超曲面的“凹凸性”概念。PI也将研究非孤立的奇点。本研究还将尝试对欧几里德空间中渐近于某些圆柱锥的8维极小超曲面进行分类，这将有助于模拟曲面向这些圆柱奇点的几何退化。本文的目的是根据最小超曲面的边界数据建立广义等周型不等式。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。

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.