Harmonic maps approach to rigidity problems

在这个项目中，PI将研究具有奇点的几何空间。大多数现有的文献在几何处理非奇异的，因此顺利，几何对象，如一个平面，一个领域，一个鞍面和他们的推广。另一方面，自然界中的大多数物体都是非光滑的，具有不同复杂程度的奇点。PI建议通过分析映射到它们以及它们之间的映射来研究某些奇异几何空间，这些映射在最小化能量的意义上是最优的。 通过对这些映射的分析，PI提出研究奇异几何空间的对称性。这样的结果在数学和物理学中是有趣的，并将导致对自然世界的更好的理解。几何空间之间的映射的能量的自然概念是通过测量映射在域的每个点处的总拉伸然后积分来定义的。调和映射是能量泛函的临界点。它们既可以看作是复分析中调和函数的推广，也可以看作是黎曼几何中参数化测地线的高维模拟。除了全测地映射，调和映射可能是将给定几何空间映射到另一个几何空间的最自然的方法。在这个提议中，PI提出将调和映射理论扩展到奇异几何，并将其应用于解决其他领域的问题。当研究调和映射时，自然会出现这样的问题：在什么条件下调和映射是正则的？如果它不是正则的，那么奇异集的Hausdorff维数是多少？奇异集是可求正的吗？我们考虑这些问题的奇异设置（即域和/或目标空间的调和映射不一定是光滑的黎曼流形）。通过更好地理解这些问题，我们认识到调和映射在奇异环境中的潜在应用，包括几何群论中的刚性问题和理解双曲结构的退化。

In this project, the PI will study geometric spaces that have singularities. Most of the existing literature in geometry deal with non-singular, thus smooth, geometric objects such as a plane, a sphere, a saddle surface and their generalizations. On the other hand, most objects in nature are non-smooth with singularities of varying degrees of complexity. The PI proposes to study certain singular geometric spaces by analyzing maps into and between them that are optimal in the sense that they minimize energy. From the analysis of these maps, the PI proposes to study symmetry properties of the singular geometric spaces. Such results are of interest in mathematics and physics and will lead to a greater understanding of the natural world.A natural notion of energy for a map between geometric spaces is defined by measuring the total stretch of the map at each point of the domain and then integrating. Harmonic maps are critical points of the energy functional. They can be seen as both a generalization of harmonic functions in complex analysis and a higher dimensional analogue of parameterized geodesics in Riemannian geometry. Next to totally geodesic maps, harmonic maps are perhaps the most natural way to map a given geometric space into another. In this proposal, the PI proposes to extend the harmonic map theory for singular geometry and apply it to solve problems in other fields. When studying harmonic maps, natural questions that arise are: Under what conditions are harmonic maps regular? If it is not regular, what is the Hausdorff dimension of the singular set? Is the singular set rectifiable? We consider these questions in the singular setting (i.e. the domain and/or target space of a harmonic map not necessarily a smooth Riemannian manifold). By gaining a better understanding of these issues, we realize the potential applications of harmonic maps in the singular setting that include rigidity problems in geometric group theory and understanding degeneration of hyperbolic structures.