Minimal Surfaces in Hyperbolic 3-Manifolds

双曲 3 流形中的最小曲面

• 批准号: 2202584
• 负责人: Baris Coskunuzer
• 金额: $ 11.49万
• 依托单位: University of Texas at Dallas
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-08-01 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2202584&HistoricalAwards=false
• 关键词: Minimal; Surfaces; Hyperbolic; Manifolds

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.

最小表面是微分几何中研究的重要对象，被认为是局部面积最小化的数学模型，它们非常特殊，并且在化学、材料科学、生物学、低维等各个领域都有应用。在广义相对论中，极小曲面作为黑洞视界的模型出现。在生物学和材料科学中，极小曲面被用于具有关键应用的材料设计中。关于最小曲面来解释一些拓扑数据分析中的基本问题在数据科学的多个领域都有强大的应用除了这项研究之外，PI 还致力于本科生和研究生的教学和培训，并通过组织研讨会、会议和写作来推进该领域的发展。特别是，该项目将研究双曲 3 流形中最小曲面的存在性和重要性质。不幸的是，双曲 3 流形是低维拓扑中最重要的流形族之一。对这些流形的拓扑理解相当具有挑战性，多年来几何分析领域一直没有考虑过这种情况下的最小曲面的研究，尽管在过去的十年中几何分析方面取得了一些突破性的成果，但双曲 3 中的最小曲面。 -流形仍然是一个未知的领域，许多基本问题仍然悬而未决。在这个项目中，PI 的目标是彻底解决无限体积双曲 3-流形中最小曲面的存在问题，并研究 Thurston 的著名之一。关于闭合双曲 3 流形中的最小叶状结构的猜想。该奖项反映了 NSF 的法定使命，并且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。