Complex asymptotics of the Möbius energy gradient of symmetric helix pairs

Abstract

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.

12 pages, 3 figures