Model of the Teichmüller space of genus zero by period maps
(English)Manuscript (preprint) (Other academic)
We define a generalized Grunsky operator corresponding to $n$ non-overlapping quasiconformally extendible conformal maps of the disk into the sphere. We show that the Grunsky operator elegantly characterizes the Dirichlet space of the sphere minus $n$ quasicircles. Furthermore, we define a map from the Teichmüller space of the sphere minus $n$ simply connected regions into the direct product of the Teichmüller space of the sphere minus $n$ points with the unit ball in the Banach space of linear maps between a direct sum of Dirichlet spaces of disks. We show that this map is holomorphic and show that it is injective up to a discrete action by a subgroup of the (quasiconformal) Teichmüller modular group. This extends the classical period mapping of compact surfaces to surfaces of genus $0$ bordered by $n$ closed curves.
Geometry Mathematical Analysis
IdentifiersURN: urn:nbn:se:uu:diva-284593OAI: oai:DiVA.org:uu-284593DiVA: diva2:920683