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SINGULARITIES OF QUADRATIC DIFFERENTIALS AND EXTREMAL TEICHMÜLLER MAPPINGS DEFINED BY DEHN TWISTS

Part of: Deformations of analytic structures Riemann surfaces

Published online by Cambridge University Press:  09 October 2009

C. ZHANG
C. ZHANG*
Affiliation:
Department of Mathematics, Morehouse College, Atlanta, GA 30314, USA (email: czhang@morehouse.edu)
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Abstract

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Let S be a Riemann surface of finite type. Let ω 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 ω can be realized as an extremal Teichmüller mapping on a surface of the same type (also denoted by S). Let ϕ be the corresponding holomorphic quadratic differential on S. We show that under certain conditions all possible nonpuncture zeros of ϕ stay away from all closures of once punctured disk components of S∖{A,B}, and the closure of each disk component of S∖{A,B} contains at most one zero of ϕ. As a consequence, we show that the number of distinct zeros and poles of ϕ is less than or equal to the number of components of S∖{A,B}.

Keywords

Riemann surfacesquasiconformal mappingsTeichmüller geodesicsabsolutely extremal mappingsTeichmüller spacesBers fiber spaces

MSC classification

Secondary: 32G15: Moduli of Riemann surfaces, Teichmüller theory 30F60: Teichmüller theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 87 , Issue 2 , October 2009 , pp. 275 - 288
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2009

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