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Inverse curvature flow of closed Legendre curves

Takashi Kagaya, Masatomo Takahashi

TL;DR

Addresses the inverse curvature flow of closed $\ell$-convex Legendre curves in the plane, extending classical flows to accommodate cusp singularities and studying global existence and long-time behavior. Builds a geometric PDE framework for $(X,\nu)$, establishes a global-in-time unique flow via reduction to a special flow with $N=\beta/\ell$, and proves the zero-set of $\beta$ is isolated with nonincreasing cardinality. Classifies self-similar solutions by solving an eigenvalue problem for the operator $\mathcal{L} f=(1/n^2)f''+f$, obtaining explicit scaling $\lambda^*(t)=e^{(1-m^2/n^2)t}$ and profile $X^*(u)$; the cusp count depends only on $(n,m)$ through a gcd formula. Using Fourier analysis of $\beta$, proves convergence, after appropriate centering and rescaling, to a self-similar profile $X^*_{n,m,a_m,b_m}$ with convergence in $C^i$, and identifies which initial Fourier modes determine the asymptotic state.

Abstract

In this paper, we deal with an inverse curvature flow of $\ell$-convex Legendre curves. Since the Legendre curve is a natural generalization of regular curve, the flow is a generalization of the classical inverse curvature flow of regular curves. For the initial value problem, we study on the unique existence of the flow in global time, the monotonicity of the number of the singular cusps with respect to t > 0 and the asymptotic behavior of the flow as $t \to \infty$. Regarding the asymptotics, the flow asymptotically converges to one of the self similar solutions by scaling appropriately, and the convergence is completely categorized depending on the initial curve.

Inverse curvature flow of closed Legendre curves

TL;DR

Addresses the inverse curvature flow of closed -convex Legendre curves in the plane, extending classical flows to accommodate cusp singularities and studying global existence and long-time behavior. Builds a geometric PDE framework for , establishes a global-in-time unique flow via reduction to a special flow with , and proves the zero-set of is isolated with nonincreasing cardinality. Classifies self-similar solutions by solving an eigenvalue problem for the operator , obtaining explicit scaling and profile ; the cusp count depends only on through a gcd formula. Using Fourier analysis of , proves convergence, after appropriate centering and rescaling, to a self-similar profile with convergence in , and identifies which initial Fourier modes determine the asymptotic state.

Abstract

In this paper, we deal with an inverse curvature flow of -convex Legendre curves. Since the Legendre curve is a natural generalization of regular curve, the flow is a generalization of the classical inverse curvature flow of regular curves. For the initial value problem, we study on the unique existence of the flow in global time, the monotonicity of the number of the singular cusps with respect to t > 0 and the asymptotic behavior of the flow as . Regarding the asymptotics, the flow asymptotically converges to one of the self similar solutions by scaling appropriately, and the convergence is completely categorized depending on the initial curve.

Paper Structure

This paper contains 6 sections, 15 theorems, 85 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Geometric equations for a flow of Legendre curves
  3. Unique existence of the inverse curvature flow
  4. Self similar solutions
  5. Asymptotic behavior
  6. Re-parametrization

Key Result

Theorem 1.1

Let $\alpha \in (0, 1)$ and $(X_0, \nu_0) \in C^{1+\alpha} (\mathbb{S}^1_{2\pi}; \mathbb{R}^2 \times \mathbb{S}^1)$ be an $\ell$-convex Legendre curve such that $X_0$ does not describe a point. Then, an inverse curvature flow $(X, \nu) \in C([0, \infty); C^1 (\mathbb{S}^1_{2\pi}; \mathbb{R}^2 \times where $(\tilde{\ell}, \tilde{\beta})$ is the Legendre curvature of $(\tilde{X}, \tilde{\nu})$ and $

Figures (5)

  • Figure 1: The figure shows the image of $X$ for $(2,3)$-cusp $(X, \nu)$ defined by $X(u) = u^2/2u^3/3$ and $\nu = \dfrac{1}{\sqrt{1+u^2}}u-1$. The Legendre curvature $(\ell, \beta)$ of this Legendre curve is $\ell(u) = 1/(1+u^2)^{3/2}, \beta(u) = u$. Therefore, the Legendre curve is $\ell$-convex and $\beta$ has a sign change at $u=0$.
  • Figure 2: All figures show the image of $X^*$ when $n=1$ and $C_2=0$. The values of the remained parameters are listed below each figure. Only the second one has a lap count of 2.
  • Figure 3: All figures show the image of $X^*$ when $n=2$ and $C_2=0$. The values of the remained parameters are listed below each figure. Only the third one has a lap count of 2 and the image of $X^*$ coincides with the first one in Figure \ref{['figure1']}.
  • Figure 4: All figures show the image of $X^*$ when $n=3$ and $C_2=0$. The values of the remained parameters are listed below each figure. Only the first one has a lap count of 2.
  • Figure 5: All figures show the image of $X^*$ when $n=1$ and $m=3$. Therefore, the number of laps for all curves is 2. To confirm that $C_1$ and $C_2$ do not affect "visual" singular cusps, the images output the curves with the values of the remained parameters listed below each figure.

Theorems & Definitions (32)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • ...and 22 more