Geometry, Analysis, and Variational Methods

几何、分析和变分方法

• 批准号: 1811840
• 负责人: Fernando Marques
• 金额: $ 42万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-08-01 至 2022-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1811840&HistoricalAwards=false
• 关键词: Geometry; Analysis; Variational; Methods

This research project concerns problems in the interface between Geometry, Analysis, and the Calculus of Variations. Questions concerning the variational theory of minimal surfaces and its applications will be investigated. Minimal surfaces are among the most natural objects in Differential Geometry. They have encountered striking applications in many other fields, like three-dimensional topology, mathematical physics, complex and conformal geometry, among others. In General Relativity minimal surfaces appear as models for the apparent horizons of black holes. The minimal surface equation plays a very important role as a model for several kinds of nonlinear phenomena in nature. Significant progress in this area has always had a great impact in mathematical analysis and the physical sciences.The research of this project will advance our basic understanding of minimal surfaces and their general existence theory. It concerns foundational questions about when these objects exist and how their properties relate to features of the ambient space. One of the goals is to develop a good understanding of the Morse-theoretic properties of the space of minimal varieties in a given Riemannian manifold. This is to be accomplished by a combination of min-max techniques and topological methods, where the relevant spaces of cycles are defined by means of Geometric Measure Theory. We will study the existence and basic properties, like the Morse index, of min-max minimal varieties.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.

该研究项目关注几何、分析和变分微积分之间的接口问题。将研究有关最小曲面变分理论及其应用的问题。最小曲面是微分几何中最自然的物体之一。它们在许多其他领域得到了引人注目的应用，例如三维拓扑、数学物理、复杂几何和共形几何等。在广义相对论中，最小表面作为黑洞视界的模型出现。最小曲面方程作为自然界多种非线性现象的模型发挥着非常重要的作用。这一领域的重大进展一直对数学分析和物理科学产生巨大影响。该项目的研究将增进我们对极小曲面及其一般存在理论的基本理解。 它涉及有关这些对象何时存在以及它们的属性如何与周围空间的特征相关的基本问题。 目标之一是更好地理解给定黎曼流形中最小簇空间的莫尔斯理论性质。这是通过最小-最大技术和拓扑方法的结合来完成的，其中循环的相关空间通过几何测度理论来定义。 我们将研究最小-最大最小变种的存在和基本属性，如莫尔斯指数。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。

项目成果

Counting minimal surfaces in negatively curved 3-manifolds

计算负弯曲 3 流形中的最小曲面

• DOI: 10.1215/00127094-2021-0057
• 发表时间: 2022
• 期刊: Duke Mathematical Journal
• 影响因子: 2.5
• 作者: Calegari, Danny;Marques, Fernando C.;Neves, André
• 通讯作者: Neves, André