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Stability of the Shrinking Semi-Circle Under the Free Boundary Curve Shortening Flow

Theodora Bourni, Nathan Burns, Mat Langford

Abstract

We establish a sharp rate of convergence for a free-boundary curve shortening flow in a convex domain in $\mathbb{R}^{2}$ which converges in finite time to a round half-point.

Stability of the Shrinking Semi-Circle Under the Free Boundary Curve Shortening Flow

Abstract

We establish a sharp rate of convergence for a free-boundary curve shortening flow in a convex domain in which converges in finite time to a round half-point.
Paper Structure (10 sections, 9 theorems, 123 equations)

This paper contains 10 sections, 9 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Linear stability
  4. The enclosed area functional
  5. The center of mass functional
  6. Evolution equations
  7. Evolution of the enclosed area functional
  8. Evolution of the center of mass functional
  9. Dynamic stability
  10. L2 decay

Key Result

Theorem 1.1

Let $\Omega \subset \mathop{\mathrm{\mathbb{R}}}\nolimits^{2}$ be a convex domain with $C^{2}$ boundary and let $\{\Gamma_{t}\}_{t \in [0,T)}$ be a maximal free boundary curve shortening flow starting from a properly embedded curve $\Gamma_{0}$ inside $\Omega$. If $T < \infty$, then $\Gamma_{t}$ con converge as $t \to T$ to the unit semi-circle in $\mathop{\mathrm{\mathbb{R}}}\nolimits^{2}_{+}$ un

Theorems & Definitions (19)

  • Theorem 1.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • ...and 9 more