平面上一类光滑凸曲线流

A kind of smooth convex curve flows in the plane

研究平面上一类严格凸曲线流(?F(u,t))/(?t)=(p(u,t)-Φ(t))N(u,t),F(u,t)是平面上的曲线族,p(u,t)是支撑函数,Φ(t)是C~∞光滑函数,N(u,t)是单位内法向量.当初始曲线F(u,0)严格凸并且Φ(t)满足适当条件时,证明了发展曲线保持凸性不变,曲线流在发展过程中具有全局存在性,且当时间t趋于无穷大时曲线在C~∞范数下收敛到有限圆.

In this paper, we consider a new kind of plane curve flows for strictly convex curves, which is defined as F（v,t）/t=（p（v,t）-φ（t））N（v,t）,where F（v,t）is a family of evolving curves with a support function p（u, t）, unit inward pointing normal vector N（u, t）, and φ（t） is a C∞ smooth function. When the initial curve F（u,0） is strictly convex and φ（t） satisfies some conditions, it is proved that the convex curve preserves convexity, and the flow exists globally, and the evolving curve converges to a circle in the C∞ metric as t → ∞.