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Scale-critical curve diffusion flows

Tatsuya Miura, Glen Wheeler

Abstract

We introduce and study a one-parameter family of curve diffusion flows with a scale-critical cubic curvature term for closed immersed planar curves. We first classify all closed stationary solutions, showing that they are precisely circles or a unique family of ``super-lemniscates''. We then analyse the dynamical stability of homothetic circles. Under a sharp spectral condition, we establish, by purely variational methods, that any small perturbation of an $ω$-fold circle monotonically approaches the unit $ω$-circle after rescaling, translation, and reparametrisation. As a corollary, we determine the sharp ranges of the parameter for the stability of an embedded circle, and of all $ω$-circles. We also uncover a striking arithmetic structure in the stability landscape, where the stability of $ω$-circles depends non-monotonically on $ω$.

Scale-critical curve diffusion flows

Abstract

We introduce and study a one-parameter family of curve diffusion flows with a scale-critical cubic curvature term for closed immersed planar curves. We first classify all closed stationary solutions, showing that they are precisely circles or a unique family of ``super-lemniscates''. We then analyse the dynamical stability of homothetic circles. Under a sharp spectral condition, we establish, by purely variational methods, that any small perturbation of an -fold circle monotonically approaches the unit -circle after rescaling, translation, and reparametrisation. As a corollary, we determine the sharp ranges of the parameter for the stability of an embedded circle, and of all -circles. We also uncover a striking arithmetic structure in the stability landscape, where the stability of -circles depends non-monotonically on .

Paper Structure

This paper contains 14 sections, 23 theorems, 174 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Classification of stationary solutions
  3. Dynamical stability of circles
  4. Classification of stationary solutions
  5. Dynamical stability of circles
  6. Length-preserving normalisation
  7. Evolution of curvature and its derivatives
  8. Decay estimate for curvature oscillation
  9. Convergence of the unnormalised flow
  10. Instability of circles
  11. Sharp parameter ranges for stability
  12. Discussion
  13. Immortal solutions and finite-time blowups
  14. Self-similar solutions

Key Result

Theorem 1.1

There exists a stationary solution to CCDF if and only if either If $c=0$, the only stationary solution is a round circle, possibly multiply covered. If $c=c_j$, the stationary solution is a unique super-lemniscate, up to similar transformations, reparametrisations, and multiple coverings. $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure 1: Super-lemniscates for $c_j=\frac{2}{(4j-1)^2}$, shown from left to right for $j=1,2,3,4,10,100$. The leftmost curve is the lemniscate of Bernoulli.
  • Figure 2: Stability region of the $\omega$-circle in the $(c,\omega)$-plane. For each integer $\omega\ge1$, the grey horizontal segment indicates the range of $c$ for which $\hat{\lambda}_{c,\omega}>0$, i.e., the $\omega$-circle is stable. The vertical dashed lines correspond to the thresholds $c=\frac{1}{9},1,\frac{3}{2}$.
  • Figure 3: Zoom of the stability diagram around $c=1.001$.

Theorems & Definitions (49)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Corollary 1.7
  • Corollary 1.8
  • Remark 1.9
  • Remark 1.10: Arithmetic stability pattern
  • ...and 39 more