Convergence of Curve Shortening Flow to Translating Soliton

TL;DR

This paper investigates the long-term behavior of convex curves under the alpha-curve shortening flow, demonstrating convergence to a unique translating soliton for a range of exponents, including the classical case.

Contribution

It establishes the convergence of convex curves with parallel line ends to a unique translating soliton under alpha-curve shortening flow for all exponents greater than 1/2.

Findings

Convex curves with parallel line ends converge to a translating soliton.

The result applies to all alpha > 1/2, including the classical case alpha=1.

The convergence is proven for complete non-compact convex curves.

Abstract

This paper concerns with the asymptotic behavior of complete non-compact convex curves embedded in $\mathbb{R}^2$ under the $\alpha$-curve shortening flow for exponents $\alpha >\frac12$. We show that any such curve having in addition its two ends asymptotic to two parallel lines, converges under $\alpha$-curve shortening flow to the unique translating soliton whose ends are asymptotic to the same parallel lines. This is a new result even in the standard case $\alpha=1$, and we prove for all exponents up to the critical case $\alpha>\frac12$.