On Moduli of Plane Domains

Abstract

It is well known that an arbitrary plane domain of finite connectivity can be mapped conformally onto an annulus minus a certain number of circular slits. The parameters defining such a canonical domain are studied in the context of Teichmüller theory. Let $\Omega$ be a plane domain bounded by $m \geqslant 3$ continua. Denote by $T(\Omega )$ the reduced Teichmüller space of $\Omega$ and by $R(\Omega )$ the space of conformal equivalence classes of domains bounded, as $\Omega$ is, by m continua. A real analytic map from $T(\Omega )$ onto an open subset $S(\Omega )$ of a $3m - 6$ dimensional product of circles and lines is constructed. It is shown that the map $T(\Omega ) \to S(\Omega )$ is a regular covering map. Finally, it is observed that there is a finite sheeted covering map $S(\Omega ) \to R(\Omega )$.

Locations

• Proceedings of the American Mathematical Society - View - PDF