On length spectrum metrics and weak metrics on Teichmüller spaces of surfaces with boundary

Abstract

Abstract. We define and study natural metrics and weak metrics on the Teichmüller space of a surface of topologically finite type with boundary. These metrics and weak metrics are associated to the hyperbolic length spectrum of simple closed curves and of properly embedded arcs in the surface. We give a comparison between the defined metrics on regions of Teichmüller space which we call ε 0 -relative -thick parts, for > 0 and ε 0 ≥ > 0. We compare the topologies defined by these metrics on Teichmüller space and we study divergence to infinity with respect to these various metrics.

Almost-isometry between Teichmüller metric and length-spectrum metric on moduli space

“…For more recent progress on the length-spectrum metric and Teichmüller metric in Teichmüller space, refer [10][11][12].…”

On metrics defined by length spectra on Teichmüller spaces of surfaces with boundary

“…The intersection of the ǫ-thick part and the ǫ 0 -relative part of T (S) is called the ǫ 0 -relative ǫ-thick part of T (S). We can deduce from [14,Theorem 3.6] that the length spectrum metric and the arc-length spectrum metric are almost-isometric on the ǫ 0 -relative ǫ-thick part of T (S). In fact, by [14,Proposition 3.5], there exists a positive constant K 0 depending on ǫ and ǫ 0 such that, for any X 1 , X 2 in the ǫ-thick ǫ 0 -relative part of T (S),…”

“…On the other hand, the "ǫ 0 -relative" upper boundedness assumption on lengths of the boundary curves is necessary for both inequality (1) and Theorem 1.6, see Example 3.8 in [14].…”

“…For surfaces with boundary, the following asymmetric metric is a natural generalization of Thurston's formula [14,2]:…”

Thurston's metric on Teichmüller space and the translation distances of mapping classes

“…This metric was introduced in [8]. It is an analogue for surfaces with boundary of the Thurston asymmetric metric [16].…”