Existence and Regularity for Variational Problems

该项目涉及各自能量功能的最佳对象，因此存在和结构结果在工程，物理和化学中都很有趣。其中最经典的研究是最小曲面，它使受固定边界约束的局部面积最小。在这个项目中特别感兴趣的是所谓的恒定平均曲率（CMC）和最小曲面以及谐波图。CMC表面对区域功能也至关重要，但现在受到封闭体积的限制。Delaunay在1841年确定了一个CMC样本家族，但又过了150年才有新的样本被发现，当时Kapouleas通过粘合技术生产了无限多的新样本。本项目研究的变分解具有许多数学领域的特征，所提出的问题和期望的结果在数学和其他领域具有广泛的兴趣。PI将继续研究几何分析中与变分问题解的存在性、规律性和紧致性有关的经典问题。该项目将使用和改进由Kapouleas开创的粘合技术，以生产最小和CMC表面的新例子。Colding-Minicozzi对完整的、适当嵌入的极小圆盘序列的奇点发展的理解，对解决螺旋面唯一性具有至关重要的意义。与磁盘完整和适当时的图像相反，具有球内边界的嵌入最小磁盘序列的奇异集结构可能是病态的。这些病理例子有助于解决唯一性和规律性的结果。粘合技术将被用于产生更疯狂的奇点，在这些环境中，通过以前的技术问题是难以解决的。对于CMC粘接，该项目旨在将欧几里得空间中开发的广义粘接技术扩展到更一般的流形。在调和映射的背景下，本课题的目的是在具有上曲率界的度量空间中建立共形调和映射的存在性。本文推广了Sacks和Uhlenbeck关于极小2球存在性的经典结果。现存地图的存在有助于回答瑟斯顿的“夸张猜想”中尚未解决的部分。

This project concerns optimal objects for their respective energy functionals, and as such existence and structural results are of interest in engineering, physics, and chemistry. The most classically studied of these are minimal surfaces, which locally minimize area subject to a fixed boundary. Of particular interest in this project are so-called constant mean curvature (CMC) and minimal surfaces as well as harmonic maps. CMC surfaces are also critical for the area functional, but with constraint now given by enclosed volume. Delaunay determined a family of CMC examples in 1841, but it was another 150 years before any new examples were known, at which time Kapouleas produced infinitely many new examples via gluing techniques. The variational solutions studied in this project have characterizations in many areas of mathematics and the proposed questions and desired results are of broad interest in mathematics and beyond.The PI will continue her study of classical questions in geometric analysis related to the existence, regularity, and compactness of solutions to variational problems. The project will use and refine the gluing techniques pioneered by Kapouleas to produce new examples of minimal and CMC surfaces. The understanding of singularity development for a sequence of complete, properly embedded minimal disks, developed by Colding-Minicozzi, was of critical importance for the resolution of the uniqueness of the helicoid. In contrast to the picture developed when the disks are complete and proper, the structure of the singular set for sequences of embedded minimal disks with boundary in a ball can be pathological. These pathological examples are helpful in the resolution of uniqueness and regularity results. Gluing techniques will be used to produce even wilder singularities in settings where problems are intractable via former techniques. For CMC gluing, the project aims to extend the generalized gluing techniques developed in Euclidean space to more general manifolds. In the setting of harmonic maps, the aim of the project is to establish the existence of conformal harmonic maps into metric spaces with upper curvature bounds. This work generalizes a classical result of Sacks and Uhlenbeck on the existence of minimal 2-spheres. Existence of the established maps could help answer the unresolved portions of Thurston's Hyperbolization Conjecture.