Geometric aspects of the curve shortening flow in the hyperbolic plane

TL;DR

This paper introduces new translation concepts in the hyperbolic plane, explicitly solves the curve shortening flow, analyzes ancient convex solutions, and provides area estimates for closed ancient solutions.

Contribution

It defines novel translations in hyperbolic geometry and explicitly solves the curve shortening flow equations for specific solution classes.

Findings

Explicit solutions for the curve shortening flow in hyperbolic plane

Characterization of ancient convex solutions with separation of variables

Area estimates for closed ancient solutions

Abstract

We define a new notion of translations in the hyperbolic plane and explicitly solve the equation of the curve shortening flow. Next, we consider the class of ancient convex solutions and solve the equation of the curve shortening flow when the curvature function is given by separation of variables. Lastly, we prove some area estimates for closed ancient solutions of the curve shortening flow.