Existence and Regularity for Variational Problems

该项目关注其各自能量泛函的最佳对象，因此存在和结构结果在工程，物理和化学中很有意义。其中最经典的研究是极小曲面，它在固定边界下局部最小化面积。在这个项目中特别感兴趣的是所谓的恒定平均曲率（CMC）和极小曲面以及调和映射。CMC曲面对于面积泛函也很重要，但现在由封闭体积给出约束。Delaunay在1841年确定了一个CMC例子家族，但在150年后才有新的例子出现，当时Kapouleas通过胶合技术产生了无限多的新例子。在这个项目中研究的变分解决方案在数学的许多领域的特点和提出的问题和预期的结果是广泛的兴趣在数学和beyond.The PI将继续她的经典问题的研究在几何分析有关的存在性，规律性，和变分问题的解决方案的紧凑性。该项目将使用和改进Kapouleas开创的胶合技术，以生产最小和CMC表面的新示例。理解奇点发展的一系列完整的，适当嵌入的最小磁盘，开发的冷Minicozzi，是至关重要的决议的唯一性的螺旋面。与当圆盘是完全的和适当的时所发展的图像相反，在球中具有边界的嵌入最小圆盘序列的奇异集的结构可以是病态的。这些病态的例子有助于解决唯一性和规律性的结果。胶合技术将被用来产生甚至更怀尔德奇异的设置中的问题是棘手的通过以前的技术。对于CMC胶合，该项目旨在将在欧几里得空间中开发的广义胶合技术扩展到更一般的流形。在调和映射的背景下，该项目的目的是建立到度量空间的共形调和映射的存在性，曲率上界。本文推广了Sacks和Uhlenbeck关于极小2-球面存在性的一个经典结果。已建立的映射的存在有助于回答瑟斯顿双曲化猜想中未解决的部分。

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.