保持弹性流嵌入性的最优阈值
Optimal thresholds for preserving embeddedness of elastic flows
AI总结：
本文研究欧几里得空间中闭曲线弹性流保持嵌入性的最优能量阈值，在余维一情形通过变分方法证明阈值形状为唯一的“弹性双泪滴”非经典曲线。
摘要：
我们考虑欧几里得空间中闭曲线的弹性流。我们获得了最优能量阈值，低于该阈值时，弹性流在所有时间内保持初始曲线的嵌入性。所获得的阈值在余维一和更高余维之间取不同的值。主要创新在于余维一的情形，其中我们得到了变分刻画：阈值形状是在所有单位旋转数的非嵌入平面闭曲线中，弯曲能量（按长度归一化）的最小化子。结果表明，最小化子唯一地由一个非经典形状给出，我们称之为“弹性双泪滴”。
英文摘要：
We consider elastic flows of closed curves in Euclidean space. We obtain optimal energy thresholds below which elastic flows preserve embeddedness of initial curves for all time. The obtained thresholds take different values between codimension one and higher. The main novelty lies in the case of codimension one, where we obtain the variational characterization that the thresholding shape is a minimizer of the bending energy (normalized by length) among all nonembedded planar closed curves of unit rotation number. It turns out that a minimizer is uniquely given by a nonclassical shape, which we call ``elastic two-teardrop''.
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