Problems in Mean Curvature Flow and Minimal Surface Theory

本课题研究两个重要的几何对象：最小曲面和平均曲率流。最小表面在数学上模拟了肥皂膜的形状；这种膜的能量与其表面积成正比，所以稳定的构型是那些面积最小的构型，即最小的表面。最小表面理论与物理学、化学、生物学和材料科学中出现的问题直接相关。更广泛地说，最小曲面是许多几何变分问题的重要模型——也就是说，人们试图找到和研究几何对象在某种意义上最优的性质的问题。除了作为物理科学的一个基本原理外，变分问题还出现在纯数学和应用数学的各个领域。相对于最小曲面是静态的，平均曲率流动是一个动态过程。粗略地说，平均曲率流以一种尽可能快地减小面积的方式不断地使表面变形。它最初是作为材料科学中某些现象的模型来研究的，也在计算机图形学和图像识别中得到了应用。此外，它与解庞卡罗猜想所用的里奇流密切相关。因此，平均曲率流在拓扑学中具有潜在的应用前景，本项目对其中的几个应用进行了探索。该项目将使用平均曲率流来研究低熵的n维欧几里得空间中的超曲面，即自然度量几何复杂性（熵）很小的超曲面。它还使用各种技术研究欧几里得三维空间中最小曲面的性质。第一个目标是更好地理解低熵超曲面的性质，特别是拓扑性质。这需要研究平均曲率流的非紧化自相似（收缩和膨胀）解的结构。总体目标是观察欧几里得四空间中的低熵超曲面是否必须平滑地约束一个闭合球。这个问题与光滑四维舍恩菲猜想密切相关，舍恩菲猜想是低维拓扑中的一个重要开放问题。该项目还研究了与最小曲面理论相关的几个问题。其中最主要的是Calabi提出的问题，由Yau提炼，并由Colding-Minicozzi部分回答，即一个完整的嵌入最小表面是否被正确嵌入。此外，该项目还探讨了射影微分几何思想与最小曲面理论之间的关系。这包括研究Korteweg-de Vries方程的模拟，以及研究球的自由边界最小表面。

This research project studies two important geometric objects: minimal surfaces and mean curvature flows. A minimal surface mathematically models the shape of a soap film; the energy of such a film is proportional to its surface area, and so stable configurations are those with least area, that is, minimal surfaces. The theory of minimal surfaces directly connects to problems arising in physics, chemistry, biology, and materials science. More broadly, minimal surfaces are an important model for many geometric variational problems -- that is, problems where one seeks to find and study the properties of geometric objects that are optimal in some sense. In addition to being a fundamental principle in the physical sciences, variational problems arise in diverse areas of pure and applied mathematics. In contrast to minimal surfaces, which are static, the mean curvature flow is a dynamic process. Roughly speaking, mean curvature flow continuously deforms a surface in a manner that decreases area as quickly as possible. It was first studied as a model of certain phenomena in materials science and has also found applications in computer graphics and image recognition. Furthermore, it is closely related to the Ricci flow that was employed in the solution of the Poincaré conjecture. As such, the mean curvature flow has promising potential applications to topology, several of which are explored by this project. This project will use the mean curvature flow to investigate hypersurfaces in n-dimensional Euclidean space of low entropy, that is, hypersurfaces for which a natural measure of geometric complexity, the entropy, is small. It also studies properties of minimal surfaces in Euclidean three-space using a variety of techniques. The first goal is to better understand properties, especially topological ones, of hypersurfaces of low entropy. This requires the investigation of the structure of non-compact self-similar (both shrinking and expanding) solutions to the mean curvature flow. The overarching objective is to see if hypersurfaces of low entropy in Euclidean four-space must smoothly bound a closed ball. This question is closely related to the smooth four-dimensional Schoenflies conjecture, an important open problem in low-dimensional topology. The project also studies several problems connected to the theory of minimal surfaces. Chief among these is the question raised by Calabi, refined by Yau, and partially answered by Colding-Minicozzi, asking whether a complete, embedded minimal surface is properly embedded. In addition, the project explores the relationship between ideas in projective differential geometry and minimal surface theory. This includes studying an analog of the Korteweg-de Vries equation and investigating free-boundary minimal surfaces in the ball.