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On the Palais-Smale condition in geometric knot theory

Nicolas Freches, Henrik Schumacher, Daniel Steenebrügge, Heiko von der Mosel

TL;DR

This work establishes the Palais-Smale condition for a broad family of nonlocal knot energies on submanifolds of arclength-parametrized curves, enabling rigorous variational analysis in geometric knot theory. By developing an abstract PS framework on submanifolds of Hilbert spaces and applying it to arclength spaces $\mathcal{A}^{1+s}$, the authors prove PS for linear combinations of the Euler-Bernoulli bending energy with energies such as $E^{\alpha,p}$, $\mathrm{TP}^{(p,q)}$, and $\mathrm{intM}^{(p,q)}$, including the tangent-point case $\mathrm{TP}^{(p,2)}$ for $p\in(4,5)$. They show existence of minimizers in ambient isotopy classes, long-time existence and subconvergence of Hilbert-gradient flows, and $C^{\infty}$-regularity of arclength-critical knots, with the crucial insight that arclength-critical points coincide with fixed-length constrained critical points under mild smoothness assumptions. The results provide a robust variational foundation for elastic knots and nonlocal knot energies, and they connect constrained criticality, symmetry, and regularity in a unified framework. The work also demonstrates that minimizers and gradient-flow dynamics respect ambient isotopy class constraints, offering tools for both theoretical analysis and potential numerical treatment of knot energies.

Abstract

We prove that various families of energies relevant in geometric knot theory satisfy the Palais-Smale condition (PS) on submanifolds of arclength para\-metrized knots. These energies include linear combinations of the Euler-Bernoulli bending energy with a wide variety of non-local knot energies, such as O'Hara's self-repulsive potentials $E^{α,p}$, generalized tangent-point energies $\TP^{(p,q)}$, and generalized integral Menger curvature functionals $\intM^{(p,q)}$. Even the tangent-point energies $\TP^{(p,2)}$ for $p\in (4,5)$ alone are shown to fulfill the (PS)-condition. For all energies mentioned we can therefore prove existence of minimizing knots in any prescribed ambient isotopy class, and we provide long-time existence of their Hilbert-gradient flows, and subconvergence to critical knots as time goes to infinity. In addition, we prove $C^\infty$-smoothness of all arclength constrained critical knots, which shows in particular that these critical knots are also critical for the energies on the larger open set of regular knots under a fixed-length constraint.

On the Palais-Smale condition in geometric knot theory

TL;DR

This work establishes the Palais-Smale condition for a broad family of nonlocal knot energies on submanifolds of arclength-parametrized curves, enabling rigorous variational analysis in geometric knot theory. By developing an abstract PS framework on submanifolds of Hilbert spaces and applying it to arclength spaces , the authors prove PS for linear combinations of the Euler-Bernoulli bending energy with energies such as , , and , including the tangent-point case for . They show existence of minimizers in ambient isotopy classes, long-time existence and subconvergence of Hilbert-gradient flows, and -regularity of arclength-critical knots, with the crucial insight that arclength-critical points coincide with fixed-length constrained critical points under mild smoothness assumptions. The results provide a robust variational foundation for elastic knots and nonlocal knot energies, and they connect constrained criticality, symmetry, and regularity in a unified framework. The work also demonstrates that minimizers and gradient-flow dynamics respect ambient isotopy class constraints, offering tools for both theoretical analysis and potential numerical treatment of knot energies.

Abstract

We prove that various families of energies relevant in geometric knot theory satisfy the Palais-Smale condition (PS) on submanifolds of arclength para\-metrized knots. These energies include linear combinations of the Euler-Bernoulli bending energy with a wide variety of non-local knot energies, such as O'Hara's self-repulsive potentials , generalized tangent-point energies , and generalized integral Menger curvature functionals . Even the tangent-point energies for alone are shown to fulfill the (PS)-condition. For all energies mentioned we can therefore prove existence of minimizing knots in any prescribed ambient isotopy class, and we provide long-time existence of their Hilbert-gradient flows, and subconvergence to critical knots as time goes to infinity. In addition, we prove -smoothness of all arclength constrained critical knots, which shows in particular that these critical knots are also critical for the energies on the larger open set of regular knots under a fixed-length constraint.
Paper Structure (17 sections, 48 theorems, 228 equations)

This paper contains 17 sections, 48 theorems, 228 equations.

Table of Contents

  1. Introduction
  2. The Palais-Smale condition
  3. Three families of knot energies on curves
  4. Main results
  5. Outline of the paper
  6. The Palais-Smale condition in an abstract setting
  7. Energies on submanifolds of Hilbert spaces
  8. Gradient flows on Riemannian manifolds
  9. The submanifold of arclength parametrized knots
  10. The arclength manifolds $\mathcal{A}^{1+s}$ and $\mathcal{A}^{1+s}_{\rm\bf z}$
  11. The topology of $\mathcal{A}^{1+s}$ and $\mathcal{A}^{1+s}_{\rm\bf z}$
  12. Critical knots in arclength submanifolds
  13. The Palais-Smale condition for bending energy and knot energies
  14. Bending energy
  15. Bending energy with self-avoidance term
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $\theta>0$, ${\rm\bf z}\in\mathbb{R}^n$, and $\mathcal{F}\in\{E^{\alpha,p},\mathop{\mathrm{TP}}\nolimits^{(p,q)}, \mathop{\mathrm{intM}}\nolimits^{(p,q)}\}$, where the respective combinations of parameters $\alpha$, $p$, and $q$ are chosen as in eq:individual-energy-spaces and eq:shrink-range. T

Theorems & Definitions (80)

  • Theorem 1.1: Bending energy plus knot energy above scale-invariance
  • Theorem 1.2: Bending energy plus Möbius energy
  • Theorem 1.3: Tangent-point energies $\mathop{\mathrm{TP}}\nolimits^{(p,2)}$
  • Corollary 1.4: Minimizing knots for total energies
  • Corollary 1.5: Minimizing knots for $\mathop{\mathrm{TP}}\nolimits^{(p,2)}$
  • Theorem 1.6: Gradient flow for total energies
  • Theorem 1.7: Gradient flow for $\mathop{\mathrm{TP}}\nolimits^{(p,2)}$
  • Theorem 1.8: Smoothness of critical points of total energies
  • Theorem 2.1: (PS)-condition for $\widetilde{\mathcal{E}}=\mathcal{E}|_\mathcal{M}$
  • proof
  • ...and 70 more