Problems in Mean Curvature Flow and Minimal Surface Theory

本研究计画研究两个重要的几何对象：极小曲面与平均曲率流。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，因此稳定的构型是那些面积最小的构型，即最小表面。极小曲面理论直接与物理学、化学、生物学和材料科学中出现的问题有关。更广泛地说，极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在某种意义上最优的几何对象的属性的问题。变分问题除了是物理科学中的一个基本原理外，还出现在纯数学和应用数学的各个领域。与静态的极小曲面不同，平均曲率流是一个动态过程。粗略地说，平均曲率流以尽可能快地减小面积的方式连续地使表面变形。它最初是作为材料科学中某些现象的模型进行研究的，并且在计算机图形学和图像识别中也有应用。此外，它与解决庞加莱猜想的里奇流密切相关。因此，平均曲率流在拓扑学中有着很好的应用前景，本项目对其中的几个应用进行了探索。本项目将使用平均曲率流来研究n维低熵欧氏空间中的超曲面，即几何复杂性的自然度量（熵）很小的超曲面。它还研究了性质的极小曲面在欧几里德三空间使用各种技术。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。这就需要研究平均曲率流的非紧自相似（收缩和膨胀）解的结构。首要的目标是看看在欧氏四维空间中的低熵超曲面是否必须光滑地约束一个闭球。这个问题与光滑四维Schoenflies猜想密切相关，后者是低维拓扑学中一个重要的公开问题。该项目还研究了与极小曲面理论有关的几个问题。其中最主要的是卡拉比提出的问题，由丘改进，部分由Colding-Minicozzi回答，询问一个完整的嵌入式极小曲面是否正确嵌入。此外，该项目还探讨了射影微分几何和最小曲面理论之间的关系。这包括研究类似的Korteweg-de弗里斯方程和调查自由边界极小曲面的球。

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.