## Journal of Mathematics of Kyoto University

### On the modulus of extremal Beltrami coefficients

#### Abstract

Let $R$ be a hyperbolic Riemann surface. Suppose the Teichmüller space $T(R)$ of $R$ is infinite-dimensional. Let $\mu$ be an extremal Beltrami coefficient on $R$ and let $[\mu ]$ be the point in $T(R)$. In this note, it is shown that if $\mu$ is not uniquely extremal, then there exists an extremal Beltrami coefficient $\nu$ in $[\mu ]$ with non-constant modulus. As an application, it follows, as is well known, that there exist infinitely many geodesics between $[\mu ]$ and the base point $[0]$ in $T(R)$ if $\mu$ is non-uniquely extremal.

#### Article information

Source
J. Math. Kyoto Univ., Volume 46, Number 2 (2006), 235-247.

Dates
First available in Project Euclid: 14 August 2009

https://projecteuclid.org/euclid.kjm/1250281774

Digital Object Identifier
doi:10.1215/kjm/1250281774

Mathematical Reviews number (MathSciNet)
MR2284341

Zentralblatt MATH identifier
1116.30013

#### Citation

Yao, Guowu; Qi, Yi. On the modulus of extremal Beltrami coefficients. J. Math. Kyoto Univ. 46 (2006), no. 2, 235--247. doi:10.1215/kjm/1250281774. https://projecteuclid.org/euclid.kjm/1250281774