Minimal Surfaces in Hyperbolic 3-Manifolds

极小曲面是在微分几何中研究的基本对象，被认为是例如肥皂膜的数学模型。由于具有局部面积最小化的特点，它们在化学、材料科学、生物学、低维拓扑和数学物理等领域有着广泛的应用。在广义相对论中，极小表面表现为黑洞视界的模型。在生物学和材料科学中，极小表面被用于具有关键应用的材料的设计。最近，PI利用著名的极小曲面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都有很强的应用。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，并通过组织研讨会、会议和撰写说明性材料来推动该领域的发展。特别是，该项目将研究双曲三维流形中极小曲面的存在性和重要性质。双曲3-流形是低维拓扑中最重要的流形之一。不幸的是，由于对这些流形的拓扑理解是相当具有挑战性的，几何分析中的极小曲面的研究多年来没有被考虑过。尽管在过去的十年中，几何分析已经取得了一些突破性的成果，但双曲三维流形中的极小曲面仍然是一个未知的领域，许多基本问题仍然是悬而未决的。在这个项目中，PI旨在完全解决无限体积双曲三维流形中极小曲面的存在问题，并研究瑟斯顿关于闭双曲三维流形中极小叶面的著名猜想之一。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 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.