由幂平均曲率和外力场之差支配的凸超曲面的发展

Evolution of hypersurfaces by powers of the mean curvature minus external force fields

考虑由幂平均曲率和外力场之差支配的超曲面的发展.证明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爆破;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 hypersurfaces moving by powers of the mean curvature minus external force fields was studied in this paper. It showed that the convexity of the hypersurface was preserved and the flow would blow up in finite time when the force fields was constant. For the linear force fields, we proved that the convexity was also preserved. And if the linear constant was negative, the flow blew up in finite time; if the linear constant was positive and the initial curvature was less than a certain constant depending on the force fields, the flow existed at any finite time interval, and expanded to infinity as the time tends to infinity; if the linear constant was positive and the initial curvature was larger than the certain constant, the flow blew up in finite time.