Harmonic maps in singular geometry

PI将继续将黎曼几何中的经典几何变分问题推广到奇异几何。主要研究了从奇异域到非正弯曲度量空间的调和映射理论。谐波映射是一种重要的解析对象，它可以用来表示与空间相关的几何、拓扑和代数对象，在几何、拓扑、代数以及数学的其他领域有许多实际和潜在的应用。在与G. Daskalopoulos的合作工作中，我们研究了多面体域上调和映射的存在性和正则性。主要应用于几何超刚性、几何群论、特征变异和霍奇理论。在与S. Yamada合作的一个项目中，我们研究了具有规定奇异点的极小曲面，补充了几何测量理论中的现有理论。这项工作包括经典高原问题的奇异版本以及等变最小曲面问题。拟议项目的主要研究对象是谐波图。地图是一个空间到另一个空间的几何表示。地图的能量有一个自然的概念用来测量总拉伸。谐波映射相对于这个能量达到平衡位置。例如，在岩石凹凸不平的表面上铺开的橡胶片的结构可以理解为地图。能量是橡胶的总张力，最小张力的构形是谐波映射。在没有外力作用的情况下，橡胶片会自然地排列成谐波图给出的位置。因此，调和映射是给定映射集合中自然和规范的选择。对这张图的性质的理解本身在数学上很有趣，但也很重要，因为它代表了自然科学中的许多重要现象。

The PI will continue the work of generalizing classical geometric variational problems in Riemannian geometry to include singular geometry. The main focus is the harmonic map theory from singular domains to non-positively curved metric spaces. Harmonic maps are important as analytical objects which can be used to represent geometric, topological and algebraic objects associated to a space and have numerous actual and potential applications in geometry, topology, and algebra, as well as other fields in mathematics. In a joint work with G. Daskalopoulos, we investigate existence and regularity properties of harmonic maps from polyhedral domains. The main applications are in geometric superrigidity, geometric group theory, character varieties and Hodge theory. In a project with S. Yamada, we study minimal surfaces with prescribed singularity complimenting the existing theory in geometric measure theory. This work includes the singular version of the classical Plateau problem as well as the equivariant minimal surface problem. The primary object of study in the proposed projects is a harmonic map. A map is a geometric representation of one space into another. There is a natural notion of energy of a map which measures the total stretch. A harmonic map achieves equilibrium position with respect to this energy. For example, a configuration of a rubber sheet spread out over a bumpy surface of a rock can be understood as a map. The energy is the total tension of the rubber and the configuration of least tension is a harmonic map. A rubber sheet, without any outside forces acting on it, will naturally arrange itself into a position given by a harmonic map. Thus, a harmonic map is a natural and canonical choice in a given set of maps. The understanding of the properties of this map is mathematically interesting in itself but also essential as it represents many important phenomena in the natural sciences.