Holomorphic families of open Riemann surfaces
It is a classical fact that the finite dimensional Teichmüller spaces can be characterized as the base spaces of families of compact Riemann surfaces that enjoy certain universal properties. In this paper we give a similar characterization of arbitrary Teichmüller spaces. We introduce a new class of ''marked families'' of (not necessarily compact) Riemann surfaces, and we show that every Teichmüller space is the base space of a marked family with an appropriate universal property. A marked family of Riemann surfaces is defined by a pair of connected complex Banach manifolds V and B and a surjective holomorphic map \(\pi\) :\(V\to B\) with special local trivializations: given any \(t_ 0\in B\) there are a Riemann surface X, an open neighborhood U of \(t_ 0\), and homeomorphisms f of \(U\times X\) onto \(\pi^{-1}(U)\) such that \(\pi(f(t,x))=t\) for all (t,x)\(\in U\times X\). These homeomorphisms f have additional properties, the most important of which is that the map \(t\mapsto f(t,x)\) from U into V is holomorphic for each x in X. They also satisfy a compatibility condition that imposes global topological conditions on \(\pi\) :\(V\to B\). The paper has three main parts. First we prove the existence of a universal marked family over any Teichmüller space. Next we prove that the compatibility condition between the local trivializations is not needed if B is simply connected. Finally, we give examples of holomorphic families of Riemann surfaces over the unit disk that are not equivalent of marked families.
holomorphic families of Riemann surfaces
scientific article; zbMATH DE number 3853454
scientific article