Dynamics of curves and surfaces governed by various geometric evolution equations

我们的研究结果总结如下：(1)证明了平面曲线上定义的某时空泛函关于曲线的随机摄动平均曲率流的变分问题存在一个临界点；(2)证明了对于有限或无限长的平面光滑非闭曲线，在任意有限时间内都存在修正总平方曲率的最陡下降流的光滑解。此外，我们还证明了对于由线段扰动的初始曲线，其解沿一定的时间序列平滑收敛到平稳解。

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.