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Constructing $λ$-Angenent curve by flow method

Pak Tung Ho

TL;DR

This work develops a flow-based construction of the $\lambda$-Angenent curve by evolving closed curves in the half-plane with metric $g=\alpha^2 g_E$ via the velocity $V_g=\frac{k_g}{K_g}$. Barriers given by lines $r=C$ and a Gauss-area framework control the flow, enabling a one-parameter family of initial curves with Gauss area $2\pi$ to yield a curve that intersects the cylinder for all time, thereby producing a closed geodesic in $(\mathbb{R}^2_+,g)$. The analysis shows convergence along suitable time sequences to a geodesic $\gamma_\infty$, which satisfies the geodesic equation in a compact region and has length strictly less than the barrier, establishing existence of a $\lambda$-Angenent curve; the approach also extends to $\lambda>1$ and connects to shrinking-doughnut structures under mean curvature flow. Overall, the paper provides a flow-based alternative to Angenent's shooting method for constructing $\lambda$-Angenent curves and situates these curves within the broader self-shrinker framework with barrier arguments and Gauss-geometry control.

Abstract

Using a modified curve shortening flow, we construct $λ$-Angenent curve, which was first constructed by the shooting method.

Constructing $λ$-Angenent curve by flow method

TL;DR

This work develops a flow-based construction of the -Angenent curve by evolving closed curves in the half-plane with metric via the velocity . Barriers given by lines and a Gauss-area framework control the flow, enabling a one-parameter family of initial curves with Gauss area to yield a curve that intersects the cylinder for all time, thereby producing a closed geodesic in . The analysis shows convergence along suitable time sequences to a geodesic , which satisfies the geodesic equation in a compact region and has length strictly less than the barrier, establishing existence of a -Angenent curve; the approach also extends to and connects to shrinking-doughnut structures under mean curvature flow. Overall, the paper provides a flow-based alternative to Angenent's shooting method for constructing -Angenent curves and situates these curves within the broader self-shrinker framework with barrier arguments and Gauss-geometry control.

Abstract

Using a modified curve shortening flow, we construct -Angenent curve, which was first constructed by the shooting method.
Paper Structure (9 sections, 17 theorems, 91 equations)

This paper contains 9 sections, 17 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. A modified curve shortening flow
  4. Evolution of lines $r=C$
  5. Short-time and long time existence
  6. A family of initial curves
  7. Convergence to a geodesic
  8. Properties of positive solutions to the graphical geodesic equation
  9. Convergence to a Shrinking Doughnut

Key Result

Theorem 1.1

There exists a simple closed geodesic $\gamma_\infty(u)=(r(u),x(u))$, $u\in\mathbb{S}^1$, in the half-plane $(\mathbb{R}^2_+,g)$. Moreover, its length $L_g(\gamma_\infty)$ with respect to the metric $g$ is less than the length of the double cover of the half-line $x=0$.

Theorems & Definitions (28)

  • Theorem 1.1
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Definition 4.1
  • Proposition 4.2
  • ...and 18 more