## Kodai Mathematical Journal

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

Erina Kinjo

#### Abstract

We consider a metric dL on the Teichmüller space T(R0) defined by the length spectrum of Riemann surfaces. H. Shiga proved that dL defines the same topology as that of the Teichmüller metric dT 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.

#### Article information

Source
Kodai Math. J., Volume 34, Number 2 (2011), 179-190.

Dates
First available in Project Euclid: 5 July 2011

https://projecteuclid.org/euclid.kmj/1309829545

Digital Object Identifier
doi:10.2996/kmj/1309829545

Mathematical Reviews number (MathSciNet)
MR2811639

Zentralblatt MATH identifier
1237.30015

#### Citation

Kinjo, Erina. On Teichmüller metric and the length spectrums of topologically infinite Riemann surfaces. Kodai Math. J. 34 (2011), no. 2, 179--190. doi:10.2996/kmj/1309829545. https://projecteuclid.org/euclid.kmj/1309829545