The local and global structure of variational solutions

本提案中的项目解决了几何分析中与特定变分问题解的存在性、结构和紧致性有关的经典问题。在之前与N.Kapouleas的工作中，PI开发了一种粘合结构，以产生欧几里德三维空间中具有有限拓扑的嵌入常平均曲率曲面的新例子。PI希望，对线性问题的适当修改将允许她扩展他们的技术，以产生无限多个新的嵌入的、常平均曲率的超曲面。作为第二个项目，PI将考虑非本征双调和映射的紧性理论。对于调和映射序列，能量量子化和正则性理论在建立紧性结果中起着关键作用。PI意图使用双调和设置的可比较结果来证明极限双调和映射和由集中点的膨胀引起的映射在图像流形中是无颈连接的。最后，PI将尝试结合和扩展她对具有有限拓扑的完备嵌入极小曲面的结构和具有边界的环形极小曲面的结构的研究，以了解具有无限拓扑的完备极小曲面的结构。变分问题的解是能量泛函的临界点，因此与物理现象密切相关。作为一个特别的例子，极小的表面是面积的关键点，在足够小的尺度上是面积最小的，肥皂膜作为自然模型出现。这个项目中考虑的对象和问题可以通过偏微分方程组、分析、变分和子流形几何的透镜来研究，对纯数学家和应用数学家以及许多物理学家都很感兴趣。

The projects in this proposal address classical questions in geometric analysis related to the existence, structure, and compactness of solutions to particular variational problems. In a previous work with N. Kapouleas, the PI developed a gluing construction to produce new examples of embedded constant mean curvature surfaces with finite topology in Euclidean three space. The PI expects that appropriate modifications to the linear problem will allow her to extend their techniques to produce infinitely many new embedded, constant mean curvature hypersurfaces. As a second project, the PI will consider compactness theory for extrinsic biharmonic maps. For sequences of harmonic maps, energy quantization and regularity theory play a key role in establishing the compactness result. The PI intends to show, using comparable results for the biharmonic setting, that the limiting biharmonic map and the maps arising from dilations at concentration points are connected in the image manifold without necks. Finally, the PI will attempt to combine and extend her study of the structure of complete, embedded minimal surfaces with finite topology and the structure of annular minimal surfaces with boundary to understand the structure of complete minimal surfaces with infinite topology. Solutions to variational problems are critical points for an energy functional and as such are deeply connected with physical phenomenon. As a particular example, minimal surfaces are critical points for area and on sufficiently small scales are area minimizers, with soap films occurring as natural models. The objects and questions considered in this project can be studied through the lens of partial differential equations, analysis, calculus of variations, and submanifold geometry and are of interest to pure and applied mathematicians as well as many physicists.