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

DOI:
https://doi.org/10.1090/S0002-9947-97-01771-6

MathSciNet review:
1390043

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Abstract: Assume both and are Riemann surfaces which are subsets of compact Riemann surfaces and respectively, and that the set has infinitely many points. We show that the only surjective complex linear isometries between the spaces of integrable holomorphic quadratic differentials on and are the ones induced by conformal homeomorphisms and complex constants of modulus 1. It follows that every biholomorphic map from the Teichmüller space of onto the Teichmüller space of is induced by some quasiconformal map of onto . 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

Email:
Nikola@math.cornell.edu

DOI:
https://doi.org/10.1090/S0002-9947-97-01771-6

Received by editor(s):
December 1, 1994

Received by editor(s) in revised form:
February 26, 1996

Article copyright:
© Copyright 1997
American Mathematical Society