论文标题
陈的曲线流的理论和数字
Theory and numerics for Chen's flow of curves
论文作者
论文摘要
在本文中，我们从神职人员和数值的角度研究陈的曲线流。我们调查了两个设置：封闭的$ω$ circles的设置，以及满足共同体条件的沉浸线。在每个设置中，我们的目标是找到几何条件，使我们能够理解流动的全局行为：对于CoCompact情况，条件是简单的，并且该论点在很大程度上是标准的。但是，对于封闭的情况，该论点非常复杂。如果没有事先变成单数，则流量将每个初始曲线缩小到一个点，我们必须确定一个条件，以确保这种行为以及确定要点以执行必要的重新缩放。我们能够成功地对曲率条件下的重新缩放进行完整分析。尽管缺乏最大和比较原理，但分析比其他四阶曲率流量（如弹性流或曲线扩散流）类似于平均曲率流的情况。我们的工作是通过对流量的数值研究来告知的，我们包括一个部分，该部分解释了所使用的算法并提供了一些进一步的模拟。
In this article we study Chen's flow of curves from theoreical and numerical perspectives. We investigate two settings: that of closed immersed $ω$-circles, and immersed lines satisfying a cocompactness condition. In each of the settings our goal is to find geometric conditions that allow us to understand the global behaviour of the flow: for the cocompact case, the condition is straightforward and the argument is largely standard. For the closed case however, the argument is quite complex. The flow shrinks every initial curve to a point if it does not become singular beforehand, and we must identify a condition to ensure this behaviour as well as identify the point in order to perform the requisite rescaling. We are able to successfully conduct a full analysis of the rescaling under a curvature condition. The analysis resembles the case of the mean curvature flow more than other fourth-order curvature flow such as the elastic flow or the curve diffusion flow, despite the lack of maximum and comparison principles. Our work is informed by a numerical study of the flow, and we include a section that explains the algorithms used and gives some further simulations.
