Applications of Analysis to Differential Geometry

本文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）函数和形式上的拉普拉斯算子的谱理论--与遍历理论的联系。ii）整体内射性定理。iii）脐点的拓扑。（4）广义希尔伯特定理。题目i）是一系列的问题来自（和有关）谱和遍历理论，动力学，消失定理和拓扑流形的非正曲率。题目ii）是我们努力理解极小曲面嵌入性的结果。它集中在寻找条件的非紧流形之间的局部同态是内射的问题。这种类型的问题出现在不同的数学领域，代数几何和数学经济学。我们这里的方法来自几何学，拓扑学和全局分析。题目iii）是微分几何中的一个经典问题，即理解脐奇点的拓扑。我们的计划是继续研究这个问题作为一个问题的爆破某些双曲型偏微分方程。题目iv）也是一个经典的问题，涉及双曲空间的等距浸入。在我们的研究方法中，从最小曲面理论的早期工作开始，我们一直寻求在技术和创造性数学之间取得平衡。这个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，双曲方程，代数和黎曼几何。我们已经对所有这些问题作出了贡献。考虑到各种各样的主题，我们愿意相信这个建议推进了数学的“内部”对话。另一方面，全球反演的工作是潜在的应用学科的兴趣，因为它解决了非线性方程组的可解性问题。

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.