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12 April 2021
 
  » arxiv » 0708.2371

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Singularities of quadratic differentials and extremal Teichm"{u}ller mappings defined by Dehn twist
Chaohui Zhang ;
Date 17 Aug 2007
AbstractLet $S$ 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 $Sackslash {A, B}$, and the closure of each disk component of $Sackslash {A, B}$ 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 $Sackslash {A, B}$.
Source arXiv, 0708.2371
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