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A new energy method for shortening and straightening complete curves

Tatsuya Miura, Fabian Rupp

TL;DR

This work introduces the direction energy framework, replacing the classical length-based energy with $D[\gamma]$ to analyze non-compact complete curves under curve shortening and higher-order geometric flows. By recasting the dynamics as gradient flows of energies that pair bending and tangent-orientation terms, it develops localized interpolation tools and global curvature controls that yield robust long-time behavior without relying on maximum principles. The authors establish a dichotomy for curve shortening flow, prove global well-posedness and precise asymptotics for fourth-order flows such as surface diffusion and Chen's flow, and characterize the asymptotic limits for the $\lambda$-elastic flow, including sharp energy thresholds and soliton rigidity. Overall, the results provide a unified, energy-based treatment of non-compact geometric evolutions in arbitrary codimensions, with implications for stability, rigidity, and asymptotic classification of solitons.

Abstract

We introduce a novel energy method that reinterprets ``curve shortening'' as ``tangent aligning''. This conceptual shift enables the variational study of infinite-length curves evolving by the curve shortening flow, as well as higher order flows such as the elastic flow, which involves not only the curve shortening but also the curve straightening effect. For the curve shortening flow, we prove convergence to a straight line under mild assumptions on the ends of the initial curve. For the elastic flow, we establish a global well-posedness theory, and investigate the precise long-time behavior of solutions. In fact, our method applies to a more general class of geometric evolution equations including the surface diffusion flow, Chen's flow, and the free elastic flow.

A new energy method for shortening and straightening complete curves

TL;DR

This work introduces the direction energy framework, replacing the classical length-based energy with to analyze non-compact complete curves under curve shortening and higher-order geometric flows. By recasting the dynamics as gradient flows of energies that pair bending and tangent-orientation terms, it develops localized interpolation tools and global curvature controls that yield robust long-time behavior without relying on maximum principles. The authors establish a dichotomy for curve shortening flow, prove global well-posedness and precise asymptotics for fourth-order flows such as surface diffusion and Chen's flow, and characterize the asymptotic limits for the -elastic flow, including sharp energy thresholds and soliton rigidity. Overall, the results provide a unified, energy-based treatment of non-compact geometric evolutions in arbitrary codimensions, with implications for stability, rigidity, and asymptotic classification of solitons.

Abstract

We introduce a novel energy method that reinterprets ``curve shortening'' as ``tangent aligning''. This conceptual shift enables the variational study of infinite-length curves evolving by the curve shortening flow, as well as higher order flows such as the elastic flow, which involves not only the curve shortening but also the curve straightening effect. For the curve shortening flow, we prove convergence to a straight line under mild assumptions on the ends of the initial curve. For the elastic flow, we establish a global well-posedness theory, and investigate the precise long-time behavior of solutions. In fact, our method applies to a more general class of geometric evolution equations including the surface diffusion flow, Chen's flow, and the free elastic flow.

Paper Structure

This paper contains 29 sections, 44 theorems, 221 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Direction energy and interpolation method
  4. Curve shortening flow
  5. Fourth-order flows of curve shortening type
  6. Fourth-order flows of curve straightening type
  7. Preliminaries
  8. Notation
  9. Basic properties of the direction energy
  10. Gradient flow structures
  11. Evolution of localized energy functionals
  12. Energy identity
  13. Localized interpolation estimates
  14. Global curvature control
  15. Evolution of localized curvature integrals
  16. ...and 14 more sections

Key Result

Theorem 2.2

Let $\gamma:[0,T)\times\mathbf{R}\to\mathbf{R}^n$ be a curve shortening flow eq:CSF with initial datum $\gamma_0\in \dot{C}^\infty(\mathbf{R};\mathbf{R}^n)$ such that $\inf_{\mathbf{R}}|\partial_x\gamma_0|>0$ and maximal existence time $T\in(0,\infty]$. Suppose that $D[\gamma_0]<\infty$. Then the di

Figures (3)

  • Figure 1: An immersed planar curve with admissible ends.
  • Figure 2: The borderline elastica, parametrized as in \ref{['eq:borderline']}.
  • Figure 3: Two loops moving apart along the elastic flow.

Theorems & Definitions (97)

  • Remark 2.1: Admissible ends
  • Theorem 2.2
  • Remark 2.3
  • Remark 2.4
  • Corollary 2.5
  • proof
  • Corollary 2.6
  • proof
  • Theorem 2.7
  • Theorem 2.8: \ref{['thm:global_lambda_elastic_flow', 'thm:convergence_lambda_elastic_flow', 'thm:convergence_elastic_flow']}
  • ...and 87 more