Soliton Solutions to the Curve Shortening Flow on the 2-dimensional hyperbolic plane

TL;DR

This paper characterizes soliton solutions to the curve shortening flow on the hyperbolic plane, linking their curvature to Minkowski space vectors, and explores their qualitative properties and classifications.

Contribution

It provides a novel characterization of solitons via Minkowski inner products and classifies all such solutions on the hyperbolic plane.

Findings

Existence of a 2-parameter family of solitons for each fixed vector.

All solitons are complete, embedded, and have curvature converging to a constant.

Each soliton is defined on the entire real line.

Abstract

We show that a curve is a soliton solution to the curve shortening flow if and only if its geodesic curvature can be written as the inner product between its tangent vector field and a fixed vector of the 3-dimensional Minkowski space. We use this characterization to provide a qualitative study of the solitons. We show that for each fixed vector there is a 2-parameter family of soliton solutions to the curve shortening flow on the 2-dimensional hyperbolic space. Moreover, we prove that each soliton is defined on the entire real line, it is embedded and its geodesic curvature converges to a constant at each end.