Hypersurfaces of Low Entropy and Mean Curvature Flow

这个项目涉及几何变分法，即研究在某种意义上对几何泛函最优的对象的性质。 这些变分问题出现在不同领域的纯数学和应用数学，也在许多物理科学。 例如，最小表面作为面积泛函的最小值出现，并提供了肥皂膜的数学模型。这个项目的重点是一个几何功能称为熵，这已经吸引了很多最近的兴趣，作为一个自然的衡量复杂的表面。用于研究这个功能的主要工具是平均曲率流，这是一个动态的过程，粗略地说，不断变形的表面的方式，尽可能快地减少其面积。 平均曲率流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，作为一种几何热流，它与Perelman用来解决庞加莱猜想的Ricci流密切相关。平均曲率流在拓扑学上也有很好的应用前景，本项目将利用平均曲率流来研究n维低熵欧氏空间中的超曲面，也就是说，几何复杂性的一个自然度量--熵很小的超曲面。 第一个目标是在L. Wang和PI，并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变形到单位球面。 这个问题是密切相关的光滑4D Schoenflies猜想-一个重要的开放问题，在低维topology.This奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

This project concerns the geometric calculus of variations, that is the study of the properties of objects which are optimal in some sense for a geometric functional. These variational problems arise in diverse areas of pure and applied mathematics and also in many physical sciences. For example, minimal surfaces arise as minima of the the area functional and provide a mathematical model of soap films. This project focuses on a geometric functional called entropy which has drawn a lot of recent interest as a natural measure of the complexity of a surface. The main tool used to study this functional is the mean curvature flow which is a dynamic process that, roughly speaking, continuously deforms a surface in a manner that decreases its area as quickly as possible. Mean curvature flow was first studied as a model of certain phenomena in materials science and has also found applications in computer graphics and image recognition. Furthermore, as a geometric heat flow, it is closely related to the Ricci flow which was used by Perelman to solve the Poincare conjecture. The mean curvature flow also has promising potential applications to topology - some of which are explored by this project.This project will use 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, entropy, is small. The first goal is to build on work of L. Wang and the PI and 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 be smoothly deformable to the unit sphere. This question is closely related to the smooth 4D Schoenflies conjecture -- an important open problem in low-dimensional topology.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.