Applications of Analysis to Differential Geometry

分析在微分几何中的应用

• 批准号: 0203637
• 负责人: Frederico Xavier
• 金额: $ 11.18万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-06-01 至 2006-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0203637&HistoricalAwards=false
• 关键词: Applications; Analysis; Differential; Geometry

ABSTRACT DMS- 0203637The object of this proposal is to continue our research on the topics below using techniques such as complex analysis, spectral theory, non-linear analysis and ergodic theory: i) Spectral theory of the Laplacian on functions and forms - connections with ergodic theory. ii) Global injectivity theorems. iii) Topology of umbilics. iv) The generalized Hilbert theorem. Topic i) is a collection of problems coming from (and relating to) spectral and ergodic theory, dynamics, vanishing theorems and the topology of manifolds of non-positive curvature. Topic ii) is an outgrowth of our efforts to understand embeddedness of minimal surfaces. It centers on the problem of finding conditions for a local diffeomorphism between non-compact manifolds to be injective.This type of question arises in areas of mathematics as diverse as algebraic geometry and mathematical economics. Our methods here come from geometry, topology and global analysis. Topic iii) is a classical problem in differential geometry namely, understanding the topology of umbilical singularities. The plan is to continue studying this problem as a question about the blow-up of certain hyperbolic partial differential equations. Topic iv) is also a classical problem, dealing with isometric immersions of hyperbolic spaces. As in iii), we plan to approach this question as a blow-up problem.In our approach to research, starting with the earlier work on minimal surface theory, we have always sought to achieve a balance between technique and creative mathematics. This proposal is full of problems and ideas coming from diverse areas, such as spectral theory, dynamical systems, hyperbolic equations, algebraic and Riemannian geometry. We have already contributed to all these questions. Given the variety of topics, we would like to believe that this proposal advances the "internal" conversation of mathematics. On the other hand, the work on Global Inversion is potentially of interest to applied disciplines, since it addresses the question of solvability of systems of non-linear equations.

DMS-0203637本建议的目的是利用复分析、谱理论、非线性分析和遍历理论等技术继续我们对以下主题的研究：i)关于函数和形式的拉普拉斯的谱理论--与遍历理论的联系。Ii)整体内射性定理。Iii)脐带的拓扑学。4)推广的希尔伯特定理。主题I)是来自(和与之相关的)谱和遍历理论、动力学、消失定理和非正曲率流形的拓扑的问题的集合。主题II)是我们理解极小曲面嵌入性的努力的结果。它的中心问题是寻找非紧流形之间的局部微分同胚是内射的条件。这类问题出现在从代数几何到数理经济学的各种数学领域。我们的方法来自几何学、拓扑学和全局分析。主题III)是微分几何中的一个经典问题，即理解脐带奇点的拓扑。我们的计划是继续把这个问题作为一个关于某些双曲型偏微分方程解爆破的问题来研究。主题IV)也是一个经典问题，涉及双曲空间的等距浸入。在我们的研究方法中，从早期关于极小曲面理论的工作开始，我们一直试图在技术和创造性数学之间取得平衡。这一建议充满了来自不同领域的问题和想法，如谱理论、动力系统、双曲型方程、代数和黎曼几何。我们已经对所有这些问题做出了贡献。考虑到话题的多样性，我们愿意相信这一提议促进了数学的“内部”对话。另一方面，关于全局逆的工作可能引起应用学科的兴趣，因为它解决了非线性方程组的可解性问题。