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.

该研究项目涉及几何，分析和变化计算之间的界面问题。将研究有关最小表面及其应用的变分理论的问题。最小表面是差异几何形状中最自然的物体之一。他们在许多其他领域都遇到了引人注目的应用，例如三维拓扑，数学物理学，复杂和共形几何形状等。总体而言，相对论最小表面是黑洞明显范围的模型。最小的表面方程在自然界中是几种非线性现象的模型起着非常重要的作用。该领域的重大进展一直在数学分析和物理科学中产生重大影响。该项目的研究将提高我们对最小表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的基本问题以及它们的性质与环境空间的特征的关系。 目标之一是对给定的riemannian歧管中最小品种空间的摩尔斯理论特性有很好的理解。这是通过最小最大技术和拓扑方法的组合来实现的，在该方法中，相关的周期空间是通过几何测量理论来定义的。 我们将研究Min-Max最小品种的Morse指数的存在和基本属性。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的审查标准，认为值得通过评估来获得支持。

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