Minimal Surfaces in Hyperbolic 3-Manifolds

极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。作为局部面积最小化，它们是非常特殊的，并且在各种领域中有应用，例如，化学、材料科学、生物学、低维拓扑学和数学物理。在广义相对论中，极小曲面是黑洞视视界的模型。在生物学和材料科学中，最小表面用于设计具有关键应用的材料。最近，PI使用关于极小曲面的众所周知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具有强大的应用。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，会议和编写临时材料来推进该领域。特别是，该项目将研究双曲3-流形中极小曲面的存在性和重要性质。双曲三维流形是低维拓扑中最重要的流形族之一。不幸的是，由于这些流形的拓扑理解是相当具有挑战性的，在这种情况下的极小曲面的研究没有被考虑的几何分析多年。尽管在过去的十年中，几何分析有了一些突破性的结果，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开放的。在这个项目中，PI的目标是彻底解决无限体积双曲三维流形中极小曲面的存在性问题，并研究Thurston关于闭双曲三维流形中极小叶理的著名定理之一。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.