Abstract
Schiffer variation of complex structure on a Riemann surfaceX 0 is achieved by punching out a parametric disc\(\bar D\) fromX 0 and replacing it by another Jordan domain whose boundary curve is a holomorphic image of\(\partial \bar D\). This change of structure depends on a complex parameter ε which determines the holomorphic mapping function around\(\partial \bar D\).
It is very natural to look for conditions under which these ε-parameters provide local coordinates for Teichmüller spaceT(X 0), (or reduced Teichmüller spaceT #(X0)). For compactX 0 this problem was first solved by Patt [8] using a complicated analysis of periods and Ahlfors' [2] τ-coordinates.
Using Gardiner's [6], [7] technique, (independently discovered by the present author), of interpreting Schiffer variation as a quasi conformal deformation of structure, we greatly simplify and generalize Patt's result. Theorems 1 and 2 below take care of all the finitedimensional Teichmüller spaces. In Theorem 3 we are able to analyse the situation for infinite dimensionalT(X 0) also. Variational formulae for the dependence of classical moduli parameters on the ε's follow painlessly.
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Nag, S. Schiffer variation of complex structure and coordinates for Teichmüller spaces. Proc. Indian Acad. Sci. (Math. Sci.) 94, 111–122 (1985). https://doi.org/10.1007/BF02880990
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DOI: https://doi.org/10.1007/BF02880990