Singularities of quadratic differentials and extremal Teichm\"{u}ller mappings defined by Dehn twists

Abstract

Let $S$ be a Riemann surface of type $(p,n)$ with $3p-3+n>0$. Let $\omega$ be a pseudo-Anosov map of $S$ that is obtained from Dehn twists along two families $\{A,B\}$ of simple closed geodesics that fill $S$. Then $\omega$ can be realized as an extremal Teichm\"{u}ller mapping on a surface of type $(p,n)$ which is also denoted by $S$. Let $\phi$ be the corresponding holomorphic quadratic differential on $S$. In this paper, we compare the locations of some distinguished points on $S$ in the $\phi$-flat metric to their locations with respect to the complete hyperbolic metric. More precisely, we show that all possible non-puncture zeros of $\phi$ must stay away from all closures of once punctured disk components of $S\backslash \{A, B\}$, and the closure of each disk component of $S\backslash \{A, B\}$ contains at most one zero of $\phi$. As a consequence of the result, we assert that the number of distinct zeros and poles of $\phi$ is less than or equal to the number of components of $S\backslash \{A, B\}$.