曲线流研究及其在凸几何中的应用

源自几何学、拓扑学、自然现象或实际问题的几何流理论已经成为现代数学的重要研究内容，它与凸体几何学有着密不可分的联系。本项目计划研究曲线流理论中一些问题及其在凸几何中的应用。具体地讲，我们将研究平面上非凸曲线的非局部几何流和空间曲线流等，重点对Gage[12]的保面积流和Jiang-Pan[21]的凸曲线流探索在初始曲线满足何种非凸条件时可以成立类似的Grayson定理。通过深入研究曲线流来探索曲线的整体几何性质与其拓扑结构之间的关系。在研究曲线流的同时挖掘与曲率流相关的几何不等式，设法证明高维欧氏空间中Green-Osher型不等式。此外，本项目的另一目的是利用支撑函数细致刻画欧氏空间中凸体的正中心集，研究（常宽）凸体的不对称性，继而帮助人们理解高维空间中的Blaschke-Lebesgue问题。

Geometric flow theory which comes from geometry, topology, natural phenomena and some practical problems has become one of the key fields of modern mathematics that bears a close relationship with convex geometry. This project plans to study some problems of curve flows and their applications in convex geometry. To be more specific, we will study the non-local geometric flows of non-convex plane curves and space curve flows and so on, try to obtain the Grayson-style theorems for Gage's area preserving flow[12] and the Jiang-Pan convex curve flow[21] under some suitable initial conditions. By deeply studying curve flows we can explore the relationship between global differential geometry properties and topological structure of curves. In the process of studying curve flows, we will study geometric inequalities related to curvature flows and try to prove the higher dimensional Green-Osher style inequalities. In addition, another purpose of this project is to use support function to describe the positive center set of convex bodies in the Euclidean space and to investigate the asymmetry of (constant width) convex bodies in order to improve the understanding of the Blaschke-Lebesgue problem in higher dimensional spaces.