[Paper Review] Sharp Entropy Bounds for Plane Curves and Dynamics of the Curve Shortening Flow

Julius Baldauf, Ao Sun|arXiv (Cornell University)|Aug 12, 2018

Geometric Analysis and Curvature Flows19 references4 citations

TL;DR

This paper establishes sharp entropy lower bounds for closed immersed plane curves under the curve shortening flow (CSF), proving that curves with turning number $m$ have entropy at least that of the $m$-covered circle if they form a type I singularity. It further constructs curves with turning number $m$ but lower entropy than the $m$-covered circle, which develop only type II singularities, and introduces a piecewise CSF that avoids unstable singularities by perturbing the flow when entropy decreases, ensuring extinction in an embedded circle or type II singularities.

ABSTRACT

We prove that a closed immersed plane curve with total curvature $2πm$ has entropy at least $m$ times the entropy of the embedded circle, as long as it generates a type I singularity under the curve shortening flow (CSF). We construct closed immersed plane curves of total curvature $2πm$ whose entropy is less than $m$ times the entropy of the embedded circle. As an application, we extend Colding-Minicozzi's notion of a generic mean curvature flow to closed immersed plane curves by constructing a piecewise CSF whose only singularities are embedded circles and type II singularities.

Motivation & Objective

• To establish sharp lower bounds on entropy for closed immersed plane curves based on their turning number and singularity type under the curve shortening flow (CSF).
• To demonstrate the existence of closed immersed curves with turning number $m$ whose entropy is strictly less than that of the $m$-covered circle, implying they cannot form type I singularities.
• To extend Colding-Minicozzi's notion of generic mean curvature flow to closed immersed plane curves by constructing a piecewise CSF that avoids entropy-unstable singularities.
• To prove that for curves with turning number greater than 1, the only possible singularities in the piecewise CSF are either embedded circles or type II singularities.

Proposed method

• Prove that if a closed immersed curve with turning number $m$ forms a type I singularity under CSF, its entropy is at least $m \cdot \lambda(\Gamma_1) = m\sqrt{2\pi/e}$, matching the entropy of the $m$-covered circle $\Gamma_m$.
• Construct explicit examples of closed immersed curves with turning number $m$ whose entropy is strictly less than $\lambda(\Gamma_m)$, showing they must form type II singularities.
• Use the concept of entropy instability: show that $\Gamma_m$ for $m \geq 2$ and all non-embedded closed shrinkers are entropy unstable, meaning small $C^\infty$ variations reduce entropy.
• Define a piecewise CSF by inserting smooth perturbations (graphs over existing curves) at critical times when entropy-unstable singularities are approached, ensuring entropy decreases at each step.
• Leverage the monotonicity of entropy under rescaling and the convergence of rescaled curves to shrinkers to show that entropy can be reduced via small perturbations near unstable singularities.
• Apply the classification of singularities for CSF of closed curves and the fact that only embedded circles can be limits of rescaling sequences for curves with turning number >1 to prove the final result on singularity types.

Experimental results

Research questions

• RQ1 What is the sharp lower bound on entropy for a closed immersed plane curve with turning number $m$ that forms a type I singularity under the curve shortening flow?
• RQ2 Can there exist closed immersed plane curves with turning number $m$ whose entropy is strictly less than that of the $m$-covered circle $\Gamma_m$?
• RQ3 Are all non-embedded closed shrinkers in the plane entropy unstable, and what is their $F$-index?
• RQ4 Can a piecewise curve shortening flow be constructed for any closed immersed plane curve such that all singularities are either embedded circles or type II singularities?
• RQ5 What is the relationship between the turning number of a curve and the type of singularity it can develop under a generic, entropy-decreasing flow?

Key findings

• For any closed immersed plane curve with turning number $m$ that forms a type I singularity under CSF, the entropy is at least $m \cdot \lambda(\Gamma_1) = m\sqrt{2\pi/e}$, matching the entropy of the $m$-covered circle $\Gamma_m$.
• There exist closed immersed curves with turning number $m$ whose entropy is strictly less than $\lambda(\Gamma_m)$, and such curves must develop only type II singularities under CSF.
• The $m$-covered circle $\Gamma_m$ is entropy unstable for all $m \geq 2$, and all non-embedded closed shrinkers in the plane are entropy unstable.
• The $F$-index of $\Gamma_m$ for $m \geq 2$ is positive, confirming the existence of at least one direction of variation that reduces entropy.
• A piecewise CSF can be constructed for any closed immersed plane curve such that the flow either becomes extinct in an embedded circle or develops only type II singularities.
• For curves with turning number greater than 1, the piecewise CSF cannot terminate in an embedded circle, as such a limit would contradict the preservation of turning number and the fact that only embedded circles can be limits of rescaling sequences for curves with $m > 1$.