Entropy in Mean Curvature Flow and Minimal Hypersurfaces

点击翻译按钮获取中文摘要

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.