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Model of the Teichmüller space of genus zero by period maps
(English)Manuscript (preprint) (Other academic)
Abstract [en]

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.

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Geometry Mathematical Analysis
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URN: urn:nbn:se:uu:diva-284593OAI: oai:DiVA.org:uu-284593DiVA: diva2:920683
Available from: 2016-04-18 Created: 2016-04-18 Last updated: 2016-09-28

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Staubach, Wolfgang
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