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Optimal thresholds for preserving embeddedness of elastic flows

Tatsuya Miura, Marius Müller, Fabian Rupp

TL;DR

The paper establishes sharp energy thresholds for preserving embeddedness of elastic flows of closed curves across codimensions. By formulating a variational problem for the scale-invariant energy $\bar{B}=LB$ on noninjective planar curves with unit rotation, the authors prove existence of a minimizer and derive a variational inequality whose analysis yields a complete classification of embedded cuspidal elasticae (ECEs) into a teardrop family and a heart-shaped family. In the planar case, the unique minimizer is the elastic two-teardrop, giving a precise threshold $C_{2T}$ that is strictly larger than the previously known $C_{8}$, while in higher codimensions the same threshold reduces to $C_{8}$, determined by the figure-eight elastica. The results translate to flow behavior: below the respective thresholds, elastic flows preserve embeddedness for all time and converge to circles (in the plane), and the thresholds are shown to be optimal by constructing embedded perturbations that develop self-intersections in finite time. The work thus links variational elasticae and geometric flow dynamics to yield exact, topology-aware energy bounds for embeddedness preservation.

Abstract

We consider elastic flows of closed curves in Euclidean space. We obtain optimal energy thresholds below which elastic flows preserve embeddedness of initial curves for all time. The obtained thresholds take different values between codimension one and higher. The main novelty lies in the case of codimension one, where we obtain the variational characterization that the thresholding shape is a minimizer of the bending energy (normalized by length) among all nonembedded planar closed curves of unit rotation number. It turns out that a minimizer is uniquely given by a nonclassical shape, which we call ``elastic two-teardrop''.

Optimal thresholds for preserving embeddedness of elastic flows

TL;DR

The paper establishes sharp energy thresholds for preserving embeddedness of elastic flows of closed curves across codimensions. By formulating a variational problem for the scale-invariant energy on noninjective planar curves with unit rotation, the authors prove existence of a minimizer and derive a variational inequality whose analysis yields a complete classification of embedded cuspidal elasticae (ECEs) into a teardrop family and a heart-shaped family. In the planar case, the unique minimizer is the elastic two-teardrop, giving a precise threshold that is strictly larger than the previously known , while in higher codimensions the same threshold reduces to , determined by the figure-eight elastica. The results translate to flow behavior: below the respective thresholds, elastic flows preserve embeddedness for all time and converge to circles (in the plane), and the thresholds are shown to be optimal by constructing embedded perturbations that develop self-intersections in finite time. The work thus links variational elasticae and geometric flow dynamics to yield exact, topology-aware energy bounds for embeddedness preservation.

Abstract

We consider elastic flows of closed curves in Euclidean space. We obtain optimal energy thresholds below which elastic flows preserve embeddedness of initial curves for all time. The obtained thresholds take different values between codimension one and higher. The main novelty lies in the case of codimension one, where we obtain the variational characterization that the thresholding shape is a minimizer of the bending energy (normalized by length) among all nonembedded planar closed curves of unit rotation number. It turns out that a minimizer is uniquely given by a nonclassical shape, which we call ``elastic two-teardrop''.

Paper Structure

This paper contains 17 sections, 33 theorems, 137 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. The minimization problem
  3. Preliminaries about Euler's elasticae
  4. Existence of minimizers and the variational inequality
  5. Self-intersection properties and regularity of solutions to the variational inequality
  6. Classification of ECEs
  7. Wavelike ECEs
  8. Orbitlike ECEs
  9. Uniqueness results for the variational inequality
  10. Consequences for the elastic flows
  11. Well-posedness of the flows
  12. Optimality of the threshold in codimension one
  13. Optimality in R^3
  14. Preservation of embeddedness
  15. Jacobi Elliptic functions
  16. ...and 2 more sections

Key Result

Theorem 1.1

If a closed smooth curve $\gamma_0:\mathbb{T}^1\to\mathbb{R}^n$ is embedded, and if then the fixed-length elastic flow (resp. $\lambda$-elastic flow) with initial datum $\gamma_0$ remains embedded for all time $t\geq0$. In addition, for any $\varepsilon>0$ (resp. $\varepsilon,\lambda>0$) there exists an embedded closed smooth curve $\gamma_0:\mathbb{T}^1\to\mathbb{R}^n$ such that and such that t

Figures (6)

  • Figure 1: Optimal configurations among nonembedded closed curves.
  • Figure 2: Embedded cuspidal elasticae (ECEs).
  • Figure 3: Elastic teardrop-hearts.
  • Figure 4: Plots of two different concatenations of $\gamma_T$ and $\gamma_H$. The rescaling ratio found in Lemma \ref{['lem:rescaling']} will yield (exactly) the left hand figure --- which has more self-intersections than predicted in Proposition \ref{['prop:selfinters']}.
  • Figure 5: Perturbation of the elastic two-teardrop, cf. Lemma \ref{['lem:existence counterexample n=2']}.
  • ...and 1 more figures

Theorems & Definitions (75)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3: LiYau1LiYau2
  • Theorem 1.4
  • Definition 2.1
  • Proposition 2.2: Planar elasticae, see e.g. LiYau1
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • ...and 65 more