Existence and Regularity for Variational Problems

这个项目涉及到各自能量泛函的最优对象，因此其存在和结构结果在工程、物理和化学中都是有意义的。其中最经典的研究是极小曲面，它局部最小化受固定边界约束的面积。在这个项目中特别感兴趣的是所谓的常值平均曲率(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。德劳奈在1841年确定了一族CMC例子，但又过了150年才知道有任何新的例子，当时卡普利亚斯通过粘合技术产生了无限多的新例子。本项目所研究的变分问题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都具有广泛的兴趣。PI将继续她对几何分析中与变分问题解的存在性、正则性和紧性有关的经典问题的研究。该项目将使用并改进由Kapouleas开创的胶合技术，以生产最小曲面和CMC曲面的新实例。由Colding-Minicozzi开发的完整的、适当嵌入的最小圆盘序列的奇异性发展的理解，对于解决螺旋面的唯一性至关重要。与当圆盘是完整且适当时所产生的图像相反，具有球中边界的嵌入极小圆盘序列的奇异集的结构可能是病态的。这些病理实例有助于解决结果的唯一性和规律性。粘合技术将被用来在通过以前的技术解决问题的环境中产生更疯狂的奇点。对于CMC粘合，该项目旨在将在欧几里得空间中开发的广义粘合技术扩展到更一般的流形。在调和映射的背景下，本课题的目的是建立具有曲率上界的度量空间中的共形调和映射的存在性。本文推广了Sack和Uhlenbeck关于极小2-球存在性的一个经典结果。已建立的地图的存在可能有助于回答瑟斯顿双曲线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.