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

Part of: Riemann surfaces Deformations of analytic structures

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

References

[1]Ahlfors, L. V. and Bers, L., ‘Riemann’s mapping theorem for variable metrics’, Ann. of Math. (2) 72 (1960), 385–404.Google Scholar
[2]Bers, L., ‘Fiber spaces over Teichmüller spaces’, Acta Math. 130 (1973), 89–126.CrossRefGoogle Scholar
[3]Bers, L., ‘An extremal problem for quasiconformal mappings and a theorem by Thurston’, Acta Math. 141 (1978), 73–98.Google Scholar
[4]Earle, C. J. and Kra, I., ‘On holomorphic mappings between Teichmüller spaces’, in: Contributions to Analysis (eds. L. V. Ahlfors et al.) (Academic Press, New York, 1974), pp. 107–124.Google Scholar
[5]Farkas, H. M. and Kra, I., Riemann Surfaces (Springer, New York and Berlin, 1980).Google Scholar
[6]Fathi, A., Laudenbach, F. and Poenaru, V., ‘Travaux de Thurston sur les surfaces’, Seminaire Orsay, Asterisque, Soc. Math. de France 66/67 (1979).Google Scholar
[7]Kra, I., ‘On the Nielsen–Thurston–Bers type of some self-maps of Riemann surfaces’, Acta Math. 146 (1981), 231–270.CrossRefGoogle Scholar
[8]Nag, S., ‘Non-geodesic discs embedded in Teichmüller spaces’, Amer. J. Math. 104 (1982), 339–408.Google Scholar
[9]Royden, H. L., ‘Automorphisms and isometries of Teichmüller space’, Ann. of Math. Stud. 66 (1971), 369–383.Google Scholar
[10]Penner, R. C., ‘Probing mapping class group using arcs’, in: Problems on Mapping Class Groups and Related Topics, Proc. Sympos. Pure Math., 74 (American Mathematical Society, Providence, RI, 2005), pp. 97–114.Google Scholar
[11]Thurston, W. P., ‘On the geometry and dynamics of diffeomorphisms of surfaces’, Bull. Amer. Math. Soc. (N.S.) 19 (1988), 417–431.CrossRefGoogle Scholar
[12]Zhang, C., ‘Commuting mapping classes and their action on the circle at infinity’, Acta Math. Sinica 52(3) (2009), 471–482.Google Scholar