Length spectra and the Teichmueller metric for surfaces with boundary

TL;DR

This paper investigates various metrics on the Teichmüller space of surfaces with boundary, comparing length spectrum-based metrics with the classical Teichmüller metric, and provides a formula relating extremal lengths of arcs to the Teichmüller metric.

Contribution

It introduces and compares new length spectrum metrics on Teichmüller space for surfaces with boundary and derives a formula linking extremal lengths of arcs to the Teichmüller metric.

Findings

Comparison of length spectrum metrics with Teichmüller metric on thick parts

Formulas for Teichmüller metric using extremal lengths of arcs

Insights into the geometry of surfaces with boundary

Abstract

We consider some metrics and weak metrics defined on the Teichmueller space of a surface of finite type with nonempty boundary, that are defined using the hyperbolic length spectrum of simple closed curves and of properly embedded arcs, and we compare these metrics and weak metrics with the Teichm\"uller metric. The comparison is on subsets of Teichm\"uller space which we call "$\epsilon_0$-relative $\epsilon$-thick parts", and whose definition depends on the choice of some positive constants $\epsilon_0$ and $\epsilon$. Meanwhile, we give a formula for the Teichm\"uller metric of a surface with boundary in terms of extremal lengths of families of arcs.