曲率流及相关的等周型几何不等式研究

几何流理论是现代数学的主流方向之一，它与微分几何学、拓扑学、凸体理论以及数学物理等有着密不可分的联系，在图像处理、火焰蔓延和医学断层扫描等实际问题中有广泛应用。本项目计划研究曲率流理论中一些问题以及与之相关的等周型几何不等式等。具体地讲，我们将研究平面曲线的非局部几何流、平面上具特定意义的crystalline曲率流、曲面上曲线流和空间曲线流等，重点对Gage[9]保面积流和Jiang-Pan[20]凸曲线流探索在初始曲线满足何种非凸条件时可以成立Grayson型定理。在研究曲线流的同时，挖掘与曲率流相关的几何不等式，尤其是等周型几何不等式，设法改进或加强Gage等周型不等式和Green-Osher等周型不等式，讨论它们的稳定性，并给出这些等周型几何不等式的离散化和它们在曲面上或高维空间中的推广形式。

Geometric flow theory has become one of the key fields of modern mathematics that bears a close relationship with differential geometry, topology, convex body theory and mathematical physics and so on, and plays important roles in image processing, flame propagation and medical tomography and other practical problems. This project plans to deal with some problems of curvature flow theory and the related isoperimetric-style geometric inequalities and geometric extreme problems. To be more specific, we will study the non-local geometric flows of plane curves, some special crystalline flows, curve flows on surfaces and space curve flows and so on, try to prove the Grayson-style theorems for Gage's area preserving flow [9] and the Jiang-Pan convex curve flow [20] under some suitable initial data. In the process of studying curve flows, we will detect geometric inequalities (isoperimertic-type inequalities in particular) related to curvature flows, try to improve or strengthen the Gage isoperimetric inequality and the Green-Osher isoperimetric inequality, discuss stability property of these inequalities and give out their discrete versions and generalizations on surfaces and in higher dimensional spaces.