Entropy in Mean Curvature Flow and Minimal Hypersurfaces

平均曲率流是超曲面在周围空间中演化的过程，使得超曲面的面积在最陡的方向上减小。极小超曲面是局部极小化面积的超曲面，它是平均曲率流的稳态解。除了本身是美丽的主题之外，平均曲率流和最小超曲面作为涉及表面张力的各种物理过程的简化模型而出现，并且它们可以应用于其他科学领域的问题，例如材料科学和计算机视觉。该项目旨在利用适当的熵的概念来研究平均曲率流和极小超曲面的全局特征。纳入该项目的重要教育活动包括：就项目中的一些问题对本科生、研究生和博士后进行辅导;招募妇女和其他代表性不足的群体;教授小型课程以吸引年轻学生到该领域;组织研讨会、讲习班和研究方案，促进年轻学者。第一个是关于平均曲率流的圆锥奇点的解析的定量理解及其在低熵欧氏空间中闭超曲面空间的高阶同伦群中的应用。第二是发展了平均曲率流自膨胀解的莫尔斯理论。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Mean curvature flow is a process that evolves hypersurfaces in an ambient space so that the area of the hypersurfaces decreases in the steepest direction. A minimal hypersurface is a hypersurface that locally minimizes the area, and it is a stationary solution to mean curvature flow. In addition to being beautiful subjects in themselves, mean curvature flow and minimal hypersurfaces arise as simplified models for various physical processes that involve surface tension, and they could be applied to questions in other scientific fields, such as materials science and computer vision. The project aims to exploit suitable notions of entropy to study global features of mean curvature flow and minimal hypersurfaces. Significant educational activities that are integrated into the project include: mentoring undergraduate and graduate students and postdocs on some questions in the project; recruiting women and other underrepresented groups; teaching mini-courses to attract young students to the field; and organizing seminars, workshops and research programs promoting young scholars.The project has three parts. The first concerns quantitative understanding of resolutions of conical singularities of mean curvature flow and its application to the higher homotopy group of the space of closed hypersurfaces in Euclidean space of low entropy. The second is to develop a Morse theory for self-expanding solutions to mean curvature flow. The last is to study geometric and topological properties of minimal hypersurfaces in sphere and hyperbolic space.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.