In this paper, we prove a general stability result for higher-order geometric flows on the circle, which basically states that if the initial condition is close to a round circle, the curve evolves smoothly and exponentially fast towards a circle (possibly not the one it started close to), and we improve on known convergence rates (which we believe are almost sharp). The polyharmonic flow is an instance of the flows to which our result can be applied. We will also present general families of flows for which our stability result applies.

中文翻译：

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圆上几何流的稳定性

在本文中，我们证明了圆上的高阶几何流的一般稳定性结果，该结果基本上表明，如果初始条件接近圆形，则曲线向圆的方向平滑且指数级地快速发展（可能不是这样）。它开始接近），并且我们提高了已知的收敛速度（我们认为这非常明显）。多谐流是可以应用我们的结果的流的一个实例。我们还将介绍适用于我们的稳定性结果的一般流量家族。