Rotational Solitons for the Curve Shortening Flow on Revolution Surfaces

TL;DR

This paper characterizes rotational solitons for the curve shortening flow on revolution surfaces in three-dimensional space, revealing their asymptotic behavior towards parallel geodesics at the ends.

Contribution

It provides a new characterization of rotational solitons on revolution surfaces and describes their asymptotic behavior, which was not previously understood.

Findings

Rotational solitons are characterized on revolution surfaces.

Open curves are asymptotic to parallel geodesics at their ends.

The behavior of these solitons is explicitly described.

Abstract

We present a characterization for the rotational soliton for the curve shortening flow (CSF) on the revolution surfaces of $\mathbb{R}^3$. Furthermore, we describe the behavior of such curves by showing that the two ends of each open curve are asymptotic to a parallel geodesic.