Horizontal curves of hyperbolic metrics

TL;DR

This paper studies how hyperbolic metrics on a fixed surface change smoothly when varied in a specific 'horizontal' direction, revealing detailed convergence properties of these metric families.

Contribution

It provides a detailed analysis of the convergence behavior of one-parameter families of hyperbolic metrics moving horizontally on a surface.

Findings

Detailed convergence properties of hyperbolic metric families

Insights into the structure of horizontal deformations

Enhanced understanding of metric variation on surfaces

Abstract

We analyse the fine convergence properties of one parameter families of hyperbolic metrics, on a fixed underlying surface, that move always in a horizontal direction, i.e. orthogonal to the action of diffeomorphisms.