Schiffer variation of complex structure and coordinates for Teichmüller spaces
Schiffer variation of complex structure on a Riemann surfaceX0 is achieved by punching out a parametric disc$$\bar D$$ fromX0 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(X0), (or reduced Teichmüller spaceT#(X0)). For compactX0 this problem was first solved by Patt  using a complicated analysis of periods and Ahlfors'  τ-coordinates.
Using Gardiner's ,  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(X0) also. Variational formulae for the dependence of classical moduli parameters on the ε's follow painlessly.
Volume 129 | Issue 5
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