平面闭曲线和欧式空间中凸曲面的曲率流研究

Differential geometry is a very important and active filed in mathematics. In recent years, curvature flows attracted more and more attention. In this project, curvature flows for smooth, closed curves embedded in a plane and convex hypersurfaces in Euclidean spaces will be concerned. . In 2007, Prof. Shing-Tung Yau proposed a problem which says that whether one can evolve a curve to another given one via a curvature flow? We generalize Prof. Gage’s area-preserving flow and hope to use it to deform one convex curve to another given one. Furthermore, we want to investigate the general case of Prof. Yau’s problem. We plan to prove Gage’s area preserving flow deforms a star-shaped curve into a circle. The gradient descent flow of the isoperimetric difference of convex hypersurfaces in Euclidean spaces will also be considered in this project. We wish to obtain new understandings for non-local curvature flows and the classical isoperimetric problem via solutions of the above problem s.

微分几何学是数学中重要而且活跃的领域。近年来，曲率流在微分几何中受到了越来越多的关注。本项目研究平面中光滑闭曲线和欧式空间中凸超曲面上的曲率流问题。. 丘成桐教授在2007年提问：如何用曲率流将平面中一条曲线演化成为另一条给定的曲线？我们推广了Gage教授的保面积流，希望能用之将一条凸曲线演化为另一条给定的凸曲线，并计划对丘教授问题的一般情形做探讨。同时我们希望能够彻底证明Gage的保面积流可以将星形曲线演化为圆周。此外本项目还拟研究欧式空间中凸超曲面等周差的梯度下降流。期望经由以上问题的解决，给具有非局部项的曲率流问题和经典等周问题带来新的理解。

作为数学中重要而且活跃的领域，微分几何学近年来受到了越来越多的关注。本项目主要研究平面中光滑闭曲线和空间中凸超曲面上的曲率流问题。具体而言主要获得了以下三个方面的结果。.首先，构造了曲率流，将平面中任一条凸曲线演化成为另一条给定的凸曲线。该工作在凸曲线的情形回答了丘成桐教授提出的公开问题。其次，对平面凸曲线的一类非局部流作了研究，特别地证明了中心对称的星形曲线在 Gage 保面积流之下演化时，曲率流能全局存在，演化曲线在有限时间内变凸以及当时间趋于无穷收敛成一个圆周。最后构造了欧氏空间中的凸超曲面的新的曲率流，证明了该曲率流将局部凸曲线演化成多重圆周，将局部凸超曲面演化成球面。