Embedded loops in the hyperbolic plane with prescribed, almost constant curvature

TL;DR

This paper investigates the existence of closed embedded curves in the hyperbolic plane with prescribed curvature close to a constant, extending understanding of geometric curves with near-constant curvature in hyperbolic geometry.

Contribution

It introduces a method to find closed embedded curves in hyperbolic space with curvature nearly constant, for small perturbations, expanding geometric analysis in hyperbolic settings.

Findings

Existence of such curves for small perturbations of curvature

Construction of curves with prescribed, almost constant curvature

Extension of classical curvature problems to hyperbolic geometry

Abstract

Given a constant $k>1$ and a real valued function $K$ on the hyperbolic plane $\mathbb H^2$, we study the problem of finding, for any $\epsilon\approx 0$, a closed and embedded curve $u^\epsilon $ in $\mathbb H^2$ having geodesic curvature $k+\epsilon K(u^\epsilon)$ at each point.