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Elastic curves and self-intersections

Tatsuya Miura

TL;DR

This work surveys the geometric and analytic theory of elastica, culminating in a sharp Li–Yau type inequality that bounds the normalized bending energy by the square of a self-intersection multiplicity. It develops explicit curvature formulas via Jacobi elliptic functions for planar and spatial elasticae and proves existence and regularity of minimizers under natural boundary conditions. The central contribution is the leaf-minimality result, establishing that the half-fold figure-eight elastica minimizes the energy among curves with a multiplicity point, enabling exact computation of the optimal constant $\varpi^*$ and informing stability questions for elastic knots and self-intersecting curves. The paper further extends the framework to $p$-elastica and elastic flow, outlining open problems on embeddedness, knot classes, and the existence of universal minimizers in broader settings.

Abstract

This is an expository note to give a brief review of classical elastica theory, mainly prepared for giving a more detailed proof of the author's Li--Yau type inequality for self-intersecting curves in Euclidean space. We also discuss some open problems in related topics.

Elastic curves and self-intersections

TL;DR

This work surveys the geometric and analytic theory of elastica, culminating in a sharp Li–Yau type inequality that bounds the normalized bending energy by the square of a self-intersection multiplicity. It develops explicit curvature formulas via Jacobi elliptic functions for planar and spatial elasticae and proves existence and regularity of minimizers under natural boundary conditions. The central contribution is the leaf-minimality result, establishing that the half-fold figure-eight elastica minimizes the energy among curves with a multiplicity point, enabling exact computation of the optimal constant and informing stability questions for elastic knots and self-intersecting curves. The paper further extends the framework to -elastica and elastic flow, outlining open problems on embeddedness, knot classes, and the existence of universal minimizers in broader settings.

Abstract

This is an expository note to give a brief review of classical elastica theory, mainly prepared for giving a more detailed proof of the author's Li--Yau type inequality for self-intersecting curves in Euclidean space. We also discuss some open problems in related topics.
Paper Structure (23 sections, 38 theorems, 157 equations, 6 figures)

This paper contains 23 sections, 38 theorems, 157 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Geometric perspective on elastica theory
  3. Bending energy
  4. Elastica: The Euler--Lagrange equation
  5. Uniqueness and dimensional rigidity
  6. Curvature equation in general dimensions
  7. Curvature equation in general dimensions
  8. Curvature equation for planar elasticae
  9. Curvature equation for spatial elasticae
  10. Explicit formula for curvature
  11. Explicit formula for planar elasticae
  12. Analytic perspective on elastica theory
  13. Existence of minimizers: Direct method
  14. Regularity
  15. Method of Lagrange multipliers
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $n,r\geq2$ be integers. Let $\gamma:\mathbf{R}/\mathbf{Z}\to\mathbf{R}^n$ be an immersed closed curve of class $W^{2,2}$ with a point of multiplicity $r$. Then In addition, equality holds if and only if $\gamma$ is a closed $r$-leafed elastica. Such a closed $r$-leafed elastica exists if and only if either $n\geq3$ or $r$ is even. If $n=2$ and $r$ is odd, then there is $\varepsilon_r>0$ such

Figures (6)

  • Figure 1: Figure-eight elastica (left) and leaf (right). Figures adapted from Miura_LiYau.
  • Figure 2: Elastic propeller: The unique closed $3$-leafed elastica Miura_LiYau.
  • Figure 3: Propeller made of unknotted wire Miura_LiYau.
  • Figure 4: Basic patterns of planar elasticae MY_2024_Crelle.
  • Figure 5: Figure-eight $p$-elasticae with $p=\frac{6}{5}$, $p=2$, $p=10$ (from left to right) MYarXiv2203.
  • ...and 1 more figures

Theorems & Definitions (84)

  • Theorem 1.1: Miura_LiYau
  • Theorem 2.1
  • proof
  • Corollary 2.2
  • Definition 2.3: Elastica
  • Theorem 2.4
  • proof
  • Corollary 2.5
  • Lemma 2.6: First variation: Geometric form
  • proof
  • ...and 74 more