Minimal Surfaces in Hyperbolic 3-Manifolds

最小曲面是微分几何中研究的基本对象，并被认为是肥皂膜等的数学模型。由于它们是局部最小化的，所以它们非常特殊，在化学、材料科学、生物学、低维拓扑和数学物理等各个领域都有应用。在广义相对论中，最小表面是黑洞视界的模型。在生物学和材料科学中，最小表面在材料设计中有着重要的应用。近年来，PI使用了众所周知的关于最小曲面的概念来解释拓扑数据分析中的一些基本问题，这些问题在数据科学的几个领域中有着强大的应用。除了这项研究，PI的目标是专注于本科生和研究生的教学和培训，以及通过组织研讨会，会议和编写说明性材料来推进该领域。特别地，本项目将研究双曲3-流形中极小曲面的存在性和重要性质。双曲型3-流形是低维拓扑中最重要的流形族之一。不幸的是，由于对这些流形的拓扑理解相当具有挑战性，因此在这种情况下对最小曲面的研究多年来一直没有被几何分析人员考虑。尽管在过去的十年里，几何分析取得了一些突破性的成果，但双曲3流形的最小曲面仍然是一个未知的领域，许多基本问题仍然没有解决。在本项目中，PI旨在彻底解决无限体积双曲3-流形中极小曲面的存在性问题，并研究Thurston关于闭双曲3-流形中极小叶的一个著名猜想。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 essential objects studied in differential geometry and considered as the mathematical model of, for example, soap films. Being locally area minimizing, they are quite special and have applications in various domains, e.g., chemistry, materials science, biology, low dimensional topology and mathematical physics. In general relativity, minimal surfaces appear as models for the apparent horizons of black holes. In biology and material science, minimal surfaces are used in the design of materials with key applications. Recently, the PI used well-known notions about minimal surfaces to explain some fundamental questions in topological data analysis which have powerful applications in several fields in data science. In addition to this research, the PI aims to focus on teaching and training of undergraduate and graduate students as well as advancing the field by organizing seminars, conferences and writing expository materials. In particular, the project will study the existence and significant properties of minimal surfaces in hyperbolic 3-manifolds. Hyperbolic 3-manifolds are one of the most important families of manifolds in the low dimensional topology. Unfortunately, as the topological understanding of these manifolds is quite challenging, the study of minimal surfaces in this setting have not been considered by those in geometric analysis for many years. Even though there are several breakthrough results in geometric analysis in the past decade, minimal surfaces in hyperbolic 3-manifolds are still an uncharted territory, and many fundamental questions are still open. In this project, the PI aims to completely resolve the existence question of minimal surfaces in infinite volume hyperbolic 3-manifolds, and study one of Thurston’s famous conjectures about the minimal foliations in closed hyperbolic 3-manifolds.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.