On Teichmüller metric and the length spectrums of topologically infinite Riemann surfaces

Abstract

We consider a metric d_L on the Teichmüller space T(R₀) defined by the length spectrum of Riemann surfaces. H. Shiga proved that d_L defines the same topology as that of the Teichmüller metric d_T on T(R₀) if a Riemann surface R₀ can be decomposed into pairs of pants such that the lengths of all their boundary components except punctures are uniformly bounded from above and below.

In this paper, we show that there exists a Riemann surface R₀ of infinite type such that R₀ cannot be decomposed into such pairs of pants, whereas the two metrics define the same topology on T(R₀). We also give a sufficient condition for these metrics to have different topologies on T(R₀), which is a generalization of a result given by Liu-Sun-Wei.