Complex methods in the asymptotics of M\"obius energy integrals of helix curves

Abstract

The M\"obius energy of a curve is a topic of interest to physical knot theorists, harmonic analysts, and geometric analysts. In particular, there are many open questions regarding the gradient flow and critical points. The Gateaux derivative indicates the variation is dependent on curvature and torsion, and so to better understand the gradient, we investigate the family of helix curves, where the ratio of torsion to curvature is a constant proportional to the pitch. We fix the radius of the helix, and study the coiling in both directions: as the helix unravels to a straight line, and as it coils infinitely tight. Specifically, we study the arclength-rescaled M\"obius energy density, which is a naturally tractable quantity under the M\"obius energy's chord-arc comparison of inverse-square laws. The asymptotics of the uncoiling helix, corresponding to an energy decay, can be proven with a short chain of estimates. However, the asymptotics of the helix as it coils infinitely tight, blowing up the energy, is a much more involved calculation. Our strategy for calculating the asymptotics, initially reminiscent of the work by Kim-Kusner, begins with a meromorphic extension of the integrand. However, proving the asymptotic equivalence requires a fundamentally distinct strategy because our integrand has infinitely many poles. keywords: M\"obius energy, helix, complex asymptotics, knot energies, physical knot theory, coiling, curves