Harmonic Maps, Minimal Surfaces, and Rigidity Problems

本文研究的是在域和/或目标空间为奇异空间的情况下谐波映射理论的发展。调和映射的这种推广的动机是它们在超刚性、几何群论、极小曲面理论和特征变异等方面的应用。PI建议通过谐波图来探索分析和群论之间的相互作用。从分析的角度来看，确定调和映射在什么条件下是正则的，如果不是正则的，则确定奇异集的结构是一个中心问题。从群论的角度来看，调和映射理论是研究刚性问题的重要工具。将使用调和映射研究的具体问题包括：NPC（非正弯曲）空间的刚性性质，复合体的二次微分和Teichmuller理论以及奇异最小曲面理论。能量的概念起源于对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，对调和映射的理解对数学家和物理学家都有很大的兴趣。调和映射很重要，因为它们是分析对象，可以用来表示与空间相关的几何、拓扑和代数对象。它们在数学的许多领域以及其他科学中有许多实际和潜在的应用。

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.