Quasiconformal Deformations of an Arbitrary Riemann Surface and Variational Formulas

2009-04-07T06:52:45Z

2009-04-07T06:52:45Z

1983-03

The purpose of this paper is to give variational formulas for Riemann's period matrices and certain kinds of meromorphic differentials on an arbitrary open Riemann surface which is deformed by quasiconformal homeomorphisms depending on a complex parameter. As the quasiconformal deformation, we consider Riemann surfaces with conformal structures decided by Beltrami differentials depending holomorphically on a complex parameter. For the sake of discussion on general open Riemann surfaces, we introduce a notion of behavior spaces in the Hilbert space of first order differential forms. The mapping induced from a quasiconformal homeomorphism preserves the behavior space. Our variational formulas are valid for the class of meromorphic differentials restricted by the behavior space. We shall show examples that each element of our period matrix is holomorphic if branch points and boundary curves vary holomorphically on a covering surface of the complex plane.

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Reprinted from the Memoirs of the Faculty of Industrial Arts, Kyoto Technical University, Science and Technology, Vol.31 (1982) pp.83-96

http://hdl.handle.net/10212/1904

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