Dynamics of curves and surfaces governed by various geometric evolution equations

我们的结果总结如下：(1)我们证明了定义在平面曲线上的一类时空泛函的变分问题存在一个临界点，它与曲线的随机摄动平均曲率流有关；(2)我们证明了对于有限或无限长的平面光滑非闭合曲线，对于任意有限时间的修正总平方曲率，存在最陡下降流的光滑解。此外，我们还证明了，对于从直线段摄动的初始曲线，其解在一定的时间序列上光滑地收敛到平稳解。

The summary of our results are stated as follows :(1)We prove that there exists a critical point of the variational problem for a certain space-time functional defined on planar curves which is concerned with stochastically perturbed mean curvature flow for curves ; (2)we show that, for a planar smooth non-closed curve with finite or infinite length, there exists a smooth solution of the steepest descent flow for the modified total squared curvature for any finite time. Moreover, we prove that, for an initial curve perturbed from line segment, the solution smoothly converges to a stationary solution along a certain sequence of time.