Curvature bound for curve shortening flow via distance comparison and a direct proof of Grayson's theorem
Curvature bound for curve shortening flow via distance comparison and a direct proof of Grayson's theorem
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DOI：
10.1515/crelle.2011.026
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发表时间：
2009-08
期刊：
影响因子：
--
通讯作者：
B. Andrews;Paul Bryan
中科院分区：
文献类型：
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作者：
B. Andrews;Paul Bryan
摘要 证明了一种新的等周估计，用于通过曲线缩短流演化的嵌入闭合曲线，归一化总长度为 2π。该估计值根据其端点之间的弧长和经过的时间来限制从下方开始的任何弦的长度。将估计应用于短段，我们直接推断出最大曲率按指数衰减到 1。这给出了格雷森定理的独立证明，不需要单调性公式或奇点分类。
Abstract 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.