The inverse curve shortening flow on the hyperbolic plane

TL;DR

This paper investigates the inverse curve shortening flow in the hyperbolic plane, classifies solitons with respect to specific vector fields, and analyzes their geometric properties and asymptotic behavior.

Contribution

It provides a classification of solitons in the hyperbolic plane related to parabolic and conformal vector fields and examines their geometric features.

Findings

Parabolic solitons are graphs on the y-axis.

Conformal solitons are graphs on the x-axis.

The paper analyzes the concavity and boundary behavior of these solitons.

Abstract

We study the inverse curve shortening flow in the hyperbolic plane $\h^2$. We classify all solitons with respect to parabolic and conformal vector fields of $\h^2$. In the upper half-plane model of $\h^2$, we prove that parabolic solitons are all graphs on the $y$-axis, whereas conformal solitons are graphs on the $x$-axis. We study the concavity of these solitons and when they approach the coordinate axes.