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Singularities of quadratic differentials and extremal Teichm"{u}ller mappings defined by Dehn twist | | Chaohui Zhang
; | | Date: |
17 Aug 2007 | | 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 $Sackslash {A, B}$, 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 | | Services: | Forum | Review | PDF | Favorites | |
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