A real variable characterization of Gromov hyperbolicity of flute surfaces

TL;DR

This paper characterizes Gromov hyperbolicity of train surfaces, including flute surfaces, using a real-valued function that captures distances between key geodesics, enabling insights into hyperbolicity stability.

Contribution

It introduces a real variable criterion for Gromov hyperbolicity of trains, broadening understanding of hyperbolic geometry in Denjoy domains.

Findings

Provides a real function criterion for hyperbolicity

Establishes stability of hyperbolicity under certain modifications

Applies to a large class of Denjoy domains

Abstract

In this paper we give a characterization of the Gromov hyperbolicity of trains (a large class of Denjoy domains which contains the flute surfaces) in terms of the behavior of a real function. This function describes somehow the distances between some remarkable geodesics in the train. This theorem has several consequences; in particular, it allows to deduce a result about stability of hyperbolicity, even though the original surface and the modified one are not quasi-isometric.