Entropy in Mean Curvature Flow and Minimal Hypersurfaces

平均曲率流是在环境空间中演化超曲面以使超曲面的面积沿最陡方向减小的过程。极小超曲面是局部最小化面积的超曲面，它是平均曲率流的定常解。平均曲率流和极小超曲面除了本身是美丽的学科外，还作为涉及表面张力的各种物理过程的简化模型而出现，它们可以应用于其他科学领域的问题，如材料科学和计算机视觉。该项目旨在利用合适的熵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指导本科生、研究生和博士后；招募妇女和其他代表性不足的群体；教授迷你课程以吸引年轻学生进入该领域；以及组织研讨会、讲习班和促进青年奖学金的研究计划。该项目由三个部分组成。第一部分是对平均曲率流的锥形奇点解的定量理解，以及它在低熵欧氏空间中闭超曲面空间的高次同伦群中的应用。二是发展了平均曲率流的自展开解的Morse理论。最后一项是研究球面和双曲空间中极小超曲面的几何和拓扑性质。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.