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.