由曲率函数和外力场之差支配的凸超曲面的发展

Evolution of Convex Hypersurfaces by Curvature Functions Minus External Force Fields

考虑由曲率函数和外力场之差支配的凸超曲面的发展.证明了外力场为常向量场时,初始超曲面的凸性是保持的,且曲率流在有限时间内爆破.对于线性外力场,初始超曲面的凸性保持.而且,若线性常数为负数,则曲率流在有限时间内收敛到一点;若线性常数为正数且初始曲率小于某一与外力场有关的常数,则曲率流光滑地存在于任意有限时间区间,并发散到无穷;若线性常数为正数且初始曲率大于某一与外力场有关的常数,则曲率流在有限时间内爆破.

Evolution of convex hypersurfaces moving by curvature functions minus external force fields is studied in this paper.It is shown that the convexity of the hypersurface is preserved and the flow will blow up in finite time when the force fields are constant.For the linear force fields,we prove that the convexity is also preserved.And if the linear constant is negative,the flow contracts to a point in finite time;if the linear constant is positive and the initial curvature is less than a certain constant depending on the force fields,the flow exists on any finite time interval,and expands to infinity as the time tends to infinity;if the Unear constant is positive and the initial curvature is larger than the certain constant,the flow blows up in finite time.