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

TL;DR

This paper investigates the asymptotic behavior of the M"obius energy of helix curves, focusing on the energy blow-up as the helix coils infinitely tightly, using complex analysis and meromorphic extensions.

Contribution

It introduces a novel approach to analyze the complex asymptotics of M"obius energy in helix curves, especially when the energy diverges.

Findings

Asymptotic decay of energy as helix unravels to a straight line.

Energy blows up as helix coils infinitely tight, with a detailed asymptotic analysis.

Use of meromorphic extension to handle infinitely many poles in the integrand.

Abstract

The M\"obius energy of a curve is a topic of interest to physical knot theorists, harmonic analysts, and geometric analysts. The Gateaux derivative indicates its variation is dependent on curvature and torsion, leading us to consider the family of helix curves, where the ratio of torsion to curvature is a constant proportional to the pitch. We fix a 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 emerges as 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 estimate. However, the asymptotics of the helix as it coils infinitely tight, blowing up the energy, is a much more involved calculation. Our strategy for proving the asymptotics, initially reminiscent of the work by Kim-Kusner, begins with a meromorphic extension of the integrand. However, proving the asymptotic equivalence is fundamentally distinct because our integrand has infinitely many poles. Much of the underlying mathematical phenomena becomes apparent only upon rigorous proof. keywords: M\"obius energy, helix, complex asymptotics, knot energies, physical knot theory, curves