# On the Convergence of the Elastic Flow in the Hyperbolic Plane

## TL;DR

This paper studies the elastic flow of closed curves in hyperbolic space, proving convergence to a minimal energy state under certain conditions and demonstrating the sharpness of energy bounds through counterexamples.

## Contribution

It establishes convergence of the elastic flow in hyperbolic space for bounded initial energy and introduces a sharp inequality to support the analysis.

## Key findings

- Convergence to the global minimizer for initial energy below 16.
- Construction of curves with infinite length blow-up.
- Sharpness of the energy bound demonstrated.

## Abstract

We examine the L^2-gradient flow of Euler's elastic energy for closed curves in hyperbolic space and prove convergence to the global minimizer for initial curves with elastic energy bounded by 16. We show the sharpness of this bound by constructing a class of curves whose lengths blow up in infinite time. The convergence results follow from a constrained sharp Reilly-type inequality.