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An isometry theorem for quadratic differentials on Riemann surfaces of finite genus

Author: Nikola Lakic
Journal: Trans. Amer. Math. Soc. 349 (1997), 2951-2967
MSC (1991): Primary 32G15; Secondary 30C62, 30C75
MathSciNet review: 1390043
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Abstract: Assume both $X$ and $Y$ are Riemann surfaces which are subsets of compact Riemann surfaces $ X_1 $ and $ Y_1, $ respectively, and that the set $ X_1 - X $ has infinitely many points. We show that the only surjective complex linear isometries between the spaces of integrable holomorphic quadratic differentials on $X$ and $Y$ are the ones induced by conformal homeomorphisms and complex constants of modulus 1. It follows that every biholomorphic map from the Teichmüller space of $X$ onto the Teichmüller space of $Y$ is induced by some quasiconformal map of $ X$ onto $Y$. Consequently we can find an uncountable set of Riemann surfaces whose Teichmüller spaces are not biholomorphically equivalent.

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Additional Information

Nikola Lakic
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853

Received by editor(s): December 1, 1994
Received by editor(s) in revised form: February 26, 1996
Article copyright: © Copyright 1997 American Mathematical Society

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