The Zoo of Solitons for Curve Shortening in $\R^n$

TL;DR

This paper classifies and describes various soliton solutions to the curve shortening flow in higher-dimensional Euclidean spaces, highlighting their geometric and asymptotic properties, including helices and spirals.

Contribution

It provides a comprehensive classification of invariant solitons for curve shortening in 1^n, including new insights into their geometric structures and asymptotic behaviors.

Findings

Identification of generalized helices as solutions

Connection between solitons and curve shortening on spheres

Most shrinking solitons are asymptotic to circles

Abstract

We provide a detailed description of solutions of Curve Shortening in $\R^n$ that are invariant under some one-parameter symmetry group of the equation, paying particular attention to geometric properties of the curves, and the asymptotic properties of their ends. We find generalized helices, and a connection with curve shortening on the unit sphere $\Sph^{n-1}$. Expanding rotating solitons turn out to be asymptotic to generalized logarithmic spirals. In terms of asymptotic properties of their ends the rotating shrinking solitons are most complicated. We find that almost all of these solitons are asymptotic to circles. Many of the curve shortening solitons we discuss here are either space curves, or evolving space curves. In addition to the figures in this paper, we have prepared a number of animations of the solitons, which can be viewed at http://www.youtube.com/user/solitons2012/videos?view=1.