对称螺旋线对的莫比乌斯能量梯度的复渐近性
Complex asymptotics of the Möbius energy gradient of symmetric helix pairs
摘要：
莫比乌斯能量是一种研究得很好的纽结能量，具有良好的规律性和自排斥性质。莫比乌斯能量梯度下的平稳曲线具有重要的理论意义，因为它们可以指示曲线在其自身力下的平衡态。在本文中，我们考虑静止对称螺旋线对下的莫比乌斯能量。通过复渐近的方法，我们描述了莫比乌斯梯度的限制行为的卷曲比趋于无穷大：梯度将发散在相反的方向取决于半径是小于还是大于$\frac{1}{2}$。最后，我们讨论了更一般的莫比乌斯高原能量的影响，其中的能量曲线，或曲线对，包括由它们界定的极小曲面的面积。对称的螺旋线对在它们之间约束螺旋面，并且应用我们的结果表明，当卷曲趋于无穷大时，固定螺旋面从下面生长到半径$\frac{1}{2}$。
英文摘要：
The Möbius energy is a well-studied knot energy with nice regularity and self-repulsive properties. Stationary curves under the Möbius energy gradient are of significant theoretical interest as they they can indicate equilibrium states of a curve under its own forces. In this paper, we consider stationary symmetric helix pairs under the Möbius energy. Through methods of complex asymptotics, we characterize the limiting behavior of the Möbius gradient as the coiling ratio tends to infinity: the gradient will diverge in opposing directions depending on whether the radius is less than or greater than $\frac{1}{2}$. We conclude by discussing the implications to the more general Möbius-Plateau energy, where the energy of a curve, or pair of curves, includes the area of the minimal surface bounded by them. Symmetric helix pairs bound a helicoid between them, and applying our result shows that stationary helicoids grow to radius $\frac{1}{2}$ from below as the coiling tends to infinity.
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