We prove the analyticity of smooth critical points for O’Hara’s knot energies ℰα,p, with p = 1 and 2 < α < 3, subject to a fixed length constraint. This implies, together with the already established regularity results for O’Hara’s knot energies, that bounded energy critical points of ℰα,1 subject to a fixed length constraint are not only C∞ but also analytic. Our approach is based on Cauchy’s method of majorants and a decomposition of the gradient that was adapted from the Möbius energy case ℰ2,1.

中文翻译：

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奥哈拉结能量临界点的规律性：从平滑到解析

我们证明了 O'Hara 结能量的平滑临界点的解析性ℰα,p， 和p = 1和2 < α < 3，受固定长度约束。这意味着，连同已经建立的 O'Hara 结能量的规律性结果，有界的能量临界点ℰα,1受固定长度约束的不仅是C∞还要分析。我们的方法是基于柯西的主要方法和从莫比乌斯能量案例改编的梯度分解ℰ2,1.