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.

这项研究项目涉及几何、分析和变分之间的接口问题。关于极小曲面的变分理论及其应用的问题将被研究。极小曲面是微分几何中最自然的对象之一。它们在许多其他领域都有惊人的应用，如三维拓扑学、数学物理、复杂几何和保角几何等。在广义相对论中，极小表面作为黑洞视界的模型出现。极小曲面方程作为自然界中多种非线性现象的模型，起着非常重要的作用。这一领域的重大进展一直对数学分析和物理科学产生重大影响，该项目的研究将促进我们对极小曲面及其一般存在理论的基本理解。它涉及这些物体何时存在以及它们的特性如何与环境空间的特征相关的基本问题。目标之一是对给定黎曼流形中极小变差空间的Morse理论性质有一个很好的理解。这将通过极小极大技术和拓扑学方法的结合来实现，其中相关的圈空间是通过几何测度论来定义的。我们将研究最小-最大最小方差的存在和基本性质，如莫尔斯指数。这一奖项反映了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。

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