Complex methods in the asymptotics of Möbius energy integrals of helix curves

Abstract: The Möbius 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 Möbius energy density, which emerges as a naturally tractable quantity under the Möbius 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: Möbius energy, helix, complex asymptotics, knot energies, physical knot theory, curves

• The paper establishes sharp asymptotics for Möbius energy density, showing decay of approximately 1/ρ² as the helix unwinds.
• It employs complex analytic techniques including infinite residue expansions and contour integration to manage infinitely many poles.
• Findings have practical implications for gradient flows in knot theory and numerical approaches to nonlocal variational problems.

Complex Methods in the Asymptotics of Möbius Energy Integrals for Helix Curves

Background: Knot Energies and Helix Geometry

The Möbius energy functional $E^{2,1}$, a specific instance of O'Hara energy functionals, is central in geometric knot theory for distinguishing between knot types and quantifying “tightness” of curves. For a curve $\gamma$, the Möbius energy is finite for embedded knots but diverges for self-intersecting curves, and its Gateaux derivative is intimately connected with curvature and torsion. Critical points of Möbius energy possess strong regularity: curves with finite Möbius energy are $C^{1,1}$, critical points are $C^\infty$, and recent results even establish real analyticity for critical points of the nonlocal operator [blatt20].

Helix curves, characterized by constant pitch $\rho$, represent a fundamental family where the ratio of torsion to curvature is constant. The rapid oscillatory geometry of helices makes them an important testbed for understanding the asymptotic behavior of nonlocal functionals such as Möbius energy. This paper fixes a helix and analyzes the Möbius energy density as the pitch varies from infinity to zero, thereby interpolating from straight lines to infinitely tight coils.

Möbius Energy Density Formulation

The Möbius energy density for a helix of pitch $\rho$ is rescaled to normalize arc length and isolate the regularization term. The helix is parametrized by $H_\rho(t) = (e^{it}, \rho t)$, and the central quantity under study is

$$I(\rho) = \sqrt{\rho^2 + 1} E(H_\rho, 0) = \int_{-\infty}^{\infty} M_\rho(t) dt$$

with

$$M_\rho(t) = \frac{\rho^2 + 1}{\rho^2 t^2 + 4 \sin^2 \frac{t}{2}} - \frac{1}{t^2}$$

for the arc-length-rescaled density.

Plots of $M_\rho(t)$ for various $\gamma$0 illustrate the behavior of the integrand as pitch varies, emphasizing its non-negativity and integrability across $\gamma$1.

Figure 1: Plots of $\gamma$2 for $\gamma$3, demonstrating integrand regularity and energy density evolution as pitch varies.

Asymptotic Regimes: Uncoiling vs Tight Coiling

Uncoiling Helix ($\gamma$4)

An elementary bounding argument yields

$\gamma$5

This corresponds to energy decay as the helix unwinds into a straight line. The proof relies on sandwich inequalities and definite integrals involving $\gamma$6 functions, confirming rapid energy decay for large pitch.

Tight Coiling ($\gamma$7)

The key result is an asymptotic blow-up:

$\gamma$8

This regime is nontrivial, requiring complex analytic methods and infinite sums over residues.

Complex Analytic Approach: Infinite Residue Expansion

The analysis leverages a meromorphic extension $\gamma$9 to the complex plane, calculating $C^{1,1}$0 via contour integration and residue calculus. Unlike torus knot energies [kusner93], which allow finite residue sums, the helix case exhibits infinitely many poles, necessitating significant technical extensions.

A detailed oriented contour $C^{1,1}$1 and its decomposition facilitate the application of the Cauchy integral formula in the presence of countably infinite singularities.

Figure 2: The oriented contour $C^{1,1}$2 and its decomposition, used to facilitate residue evaluation in the upper complex half-plane.

Residues are located at zeros of the exponential polynomial

$C^{1,1}$3

uniquely in vertical half-strips of the complex plane. The real and imaginary parts of these roots grow approximately linearly and logarithmically, respectively, with $C^{1,1}$4.

Numerical evidence supports the distribution of poles and their bifurcation as pitch crosses a critical threshold.

Figure 3: Illustration of pole bifurcation and root behavior for $C^{1,1}$5, $C^{1,1}$6, and $C^{1,1}$7, highlighting the transition in root structure.

Rigorous Pole Localization and Approximation

The existence and uniqueness of roots in each vertical strip are proven via an intermediate value theorem (the X-Principle), supported by detailed analysis of transcendental quantities such as solutions to $C^{1,1}$8. The poles $C^{1,1}$9 lack closed forms, but are closely approximated by

$C^\infty$0

The error

$C^\infty$1

is controlled uniformly for small $C^\infty$2, enabling the use of approximate residues in asymptotic expansions.

Asymptotic Expansion and Infinite Series Analysis

Summing over residues yields

$C^\infty$3

with the corresponding approximation

$C^\infty$4

Euler-Maclaurin arguments show

$C^\infty$5

as $C^\infty$6. Proving the asymptotic equivalence $C^\infty$7 requires precise error control and the “Transfer Theorem" connecting asymptotics of complex series to their real-valued imaginary parts.

Significance and Mathematical Implications

The analysis demonstrates the Möbius energy density's monotonicity and strict decrease in pitch, with no critical values analogous to torus knot cases. The rigorous expansion of $C^\infty$8 into a series of residues and its approximation enables computation of energy blow-up rates for tightly coiled helices. The presence of infinitely many poles is handled by sophisticated complex analytic arguments, highlighting subtle features in the interplay between geometry and harmonic analysis.

The work also clarifies the role of real-analyticity and convergence in nonlocal functionals, extending prior results in physical knot theory, geometric analysis, and the broader theory of exponential polynomials [hitw23].

Theoretical and Practical Implications

Understanding Möbius energy scaling for helices informs gradient flows and minimization schemes for knot energies, reinforcing the geometric intuition that unwinding local “helical” regions lowers energy rapidly, while energy blows up in the limit of tight coiling. The precise asymptotics support numerical approaches and further regularity results in the study of nonlocal PDEs and geometric variational principles.

These methods and their technical apparatus—complexification, residue calculus, pole localization via transcendental equations—are adaptable to other classes of curves with oscillatory or periodic structure, including applications in spectral geometry and nonlinear analysis.

Conclusion

This paper rigorously analyzes the Möbius energy density of helix curves via complex analytic and asymptotic methods. It establishes sharp decay as pitch increases and a logarithmic blow-up as pitch decreases, furnishing asymptotic equivalences through residue expansions and error-controlled approximations. The mathematical machinery developed is broadly relevant for intricate questions in knot energies and geometric analysis, setting the stage for further study of nonlocal functionals in oscillatory geometric settings (2605.12815).