Schiffer variation of complex structure and coordinates for Teichmuller spaces

Abstract

Schiffer variation of complex structure on a Riemann surface X₀ is achieved by punching out a parametric disc D̅ from X₀ and replacing it by another Jordan domain whose boundary curve is a holomorphic image of ∂D̅. This change of structure depends on a complex parameter e which determines the holomorphic mapping function around. It is very natural to look for conditions under which these ε-parameters provide local coordinates for Teichmuller space T(X₀), (or reduced Teichmuller space T^#(X₀)). For compact X₀ 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 Teichmuller spaces. In Theorem 3 we are able to analyse the situation for infinite dimensional T(X₀) also. Variational formulae for the dependence of classical moduli parameters on the ε's follow painlessly.