Faber and grunsky operators on bordered Riemann surfaces of arbitrary genus and the Schiffer isomorphism
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Date
2020-04-13
Authors
Shirazi, Mohammad
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Abstract
Let Sigma be a bordered Riemann surface of genus g > 0 with n>1 borders Gamma_1, ..., Gamma_n each one homeomorphic to the unit circle. This surface can be described as a compact Riemann surface R of the same genus with finitely many simply connected domains Omega_1, ..., Omega_n removed. Let f_k be a conformal map from the unit disc D in C onto Omega_k, k =1,..., n. We first generalize the classical Faber and Grunsky operators, to operators on Sigma associated to the maps f = (f_1, ..., f_n). These two operators are used to characterize the holomorphic Dirichlet space D(Sigma). More precisely, we show that the pull-back of the conformally-non-tangential boundary values of functions in D(Sigma) under f is the graph of the Grunsky operator. We also show that the Grunsky operator is a bounded operator of norm strictly less than one. So far this had only be proven for the genus zero case. The central problem is to prove that the Faber operator is a bounded isomorphism. This is done by using the Schiffer operators, which we generalize it for Sigma in this work. We characterize the function space on which the Schiffer operator is a bounded isomorphism. This characterization depends on the topology of the surface. The condition that the boundary curves are quasicircles plays a vital role in this proof. This is also a generalization of the genus zero case. The Grunsky operator is used to define a map, say Pi_g, on the Teichmüller space of Sigma. The map Pi_g has some analogies with the classical period map defined for compact surfaces. We conclude the thesis with a conjecture regarding the holomorphicity of this map for g, which was an important source of motivation for the results of the thesis.
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Teichmüller Theory, mathematics, complex analysis, Riemann surface