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Harmonic Maps, Minimal Surfaces, and Rigidity Problems

调和图、最小曲面和刚度问题

基本信息

批准号：
1105599
负责人：
Chikako Mese
金额：
$ 18.55万
依托单位：
Johns Hopkins University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2011
资助国家：
美国
起止时间：
2011-08-15 至 2015-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1105599&HistoricalAwards=false
关键词：
Harmonic Maps Minimal Surfaces Rigidity

项目摘要

The proposed research concerns the development of harmonic map theory in the setting when the domain and/or the target spaces are singular spaces. The motivation of such an extension of harmonic maps is for their application in superrigidity, geometric group theory, minimal surface theory and character varieties. The PI proposes to explore the interplay between analysis and group theory via harmonic maps. From the analysis point of view, it is a central problem to determine under what conditions harmonic maps are regular and, if not, to determine the structure of the singular sets. From a group theory point of view, harmonic map theory is an important tool in studying rigidity questions. Among the specific problem that will be investigated using harmonic maps are: rigidity properties of NPC (non-positively curved) spaces, quadratic differentials and Teichmuller theory of complexes and the theory of singular minimal surfaces. The notion of energy has its origins in the mathematical description of the physical world. Roughly speaking, the energy of a map between two spaces measures the total stretch of the map, and harmonic maps are the critical points of the energy functional. Thus, the understanding of harmonic maps is of great interest to both mathematicians and physicists. Harmonic maps are important because they are analytical objects that can be used to represent geometric, topological and algebraic objects associated to a space. They have numerous actual and potential applications in many fields of mathematics or well as in other sciences.

所提出的研究关注的调和映射理论的发展时，域和/或目标空间是奇异空间的设置。调和映射的这种推广的动机是为了它们在超刚性、几何群论、极小曲面理论和特征标簇中的应用。 PI建议通过调和映射来探索分析和群论之间的相互作用。从分析的观点来看，确定调和映射在什么条件下是正则的，如果不是正则的，确定奇异集的结构是一个中心问题。从群论的观点来看，调和映射理论是研究刚性问题的一个重要工具。其中具体的问题，将调查使用调和映射是：刚性性质的NPC（非积极弯曲）空间，二次微分和Teichmuller理论的复杂性和理论的奇异极小曲面。能量的概念起源于物理世界的数学描述。粗略地说，两个空间之间的映射的能量度量了映射的总拉伸，调和映射是能量泛函的临界点。因此，对调和映射的理解是数学家和物理学家都非常感兴趣的。调和映射很重要，因为它们是可用于表示与空间关联的几何、拓扑和代数对象的分析对象。它们在数学或其他科学的许多领域都有许多实际和潜在的应用。