## Kodai Mathematical Journal

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

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

#### Abstract

We consider a metric *d*_{L} on the Teichmüller space *T*(*R*_{0}) 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*(*R*_{0}) if a Riemann surface *R*_{0} 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 *R*_{0} of infinite type such that *R*_{0} cannot be decomposed into such pairs of pants, whereas the two metrics define the same topology on *T*(*R*_{0}). We also give a sufficient condition for these metrics to have different topologies on *T*(*R*_{0}), 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

**Permanent link to this document**

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