Geometry, analysis and variational methods

几何、分析和变分方法

• 批准号: 1509027
• 负责人: Fernando Marques
• 金额: $ 37.1万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1509027&HistoricalAwards=false
• 关键词: Geometry; analysis; variational; methods

The PI investigates questions concerning the variational theory of minimal submanifolds and its applications. 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. This research project contains a number of problems at the interface between geometry, analysis and the calculus of variations. One of the project's 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. The PI will study the existence and basic properties, like the Morse index, of min-max minimal varieties.

PI研究有关极小子流形的变分理论及其应用的问题。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，如三维拓扑学、数学物理、复几何和共形几何等，都有惊人的应用。在广义相对论中，极小曲面是黑洞视视界的模型。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对数学分析和物理科学产生了巨大的影响。该项目的研究将促进我们对极小曲面及其一般存在性理论的基本理解。它涉及的基本问题是这些物体何时存在，以及它们的属性如何与周围空间的特征相关。这个研究项目包含了一些问题之间的接口几何，分析和变分法。该项目的目标之一是发展一个很好的理解的莫尔斯理论性质的空间的最小品种在一个给定的黎曼流形。这是要完成的最小-最大技术和拓扑方法的组合，其中相关的空间的周期定义的几何测度理论的手段。PI将研究最小-最大极小簇的存在性和基本性质，如莫尔斯指数。