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

TL;DR

This paper proves a precise rate at which a free-boundary curve shortening flow in a convex domain converges to a round half-point, demonstrating stability and finite-time convergence.

Contribution

It establishes a sharp convergence rate for the free-boundary curve shortening flow in convex domains, advancing understanding of flow stability and convergence behavior.

Findings

Convergence occurs in finite time to a round half-point.

The rate of convergence is sharp and explicitly characterized.

Flow stability under the free boundary condition is confirmed.

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.