Curve-shortening flow of open, elastic curves in $\mathbb{R}^2$ with repelling endpoints: A minimizing movement approach

The Dirichlet problem for curve shortening flow

We investigate the evolution of open curves with fixed endpoints under the curve shortening flow, which evolves curves in proportion to their curvature. Using a distance comparison of Huisken, we determine the long-term behavior of open curves with fixed endpoints evolving in certain convex domains on surfaces of constant curvature. Specifically, we show that such curves do not develop singularities, and evolve to geodesics.

Curvature bound for curve shortening flow via distance comparison and a direct proof of Grayson's theorem

A new isoperimetric estimate is proved for embedded closed curves evolving by curve shortening flow, normalized to have total length $2π$. The estimate bounds the length of any chord from below in terms of the arc length between its endpoints and elapsed time. Applying the estimate to short segments we deduce directly that the maximum curvature decays exponentially to 1. This gives a self-contained proof of Grayson's theorem which does not require the monotonicity formula or the classification of singularities.

平面网络各向异性曲线缩短流的稳定性分析

Stability analysis for the anisotropic curve shortening flow of planar networks

该文研究三曲线平面网络的各向异性曲线缩短流，通过证明各向异性曲率能量满足Łojasiewicz-Simon梯度不等式，得出接近局部能量极小值的初始数据对应流全局存在且收敛于能量极小值的稳定性结论。