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Extremal problems for quasiconformal maps of punctured plane domains

Author: Vladimir Markovic
Journal: Trans. Amer. Math. Soc. 354 (2002), 1631-1650
MSC (1991): Primary 41A65, 30C75
Published electronically: November 19, 2001
MathSciNet review: 1873021
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Abstract: The main goal of this paper is to give an affirmative answer to the long-standing conjecture which asserts that the affine map is a uniquely extremal quasiconformal map in the Teichmüller space of the complex plane punctured at the integer lattice points. In addition we derive a corollary related to the geometry of the corresponding Teichmüller space. Besides that we consider the classical dual extremal problem which naturally arises in the tangent space of the Teichmüller space. In particular we prove the uniqueness of Hahn-Banach extension of the associated linear functional given on the Bergman space of the integer lattice domain. Several useful estimates related to the local and global properties of integrable meromorphic functions and the delta functional (see the definition below) are also obtained. These estimates are intended to study the behavior of integrable functions near singularities and they are valid in general settings.

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Additional Information

Vladimir Markovic
Affiliation: Institute of Mathematics, University of Warwick, Coventry, CV4 7AL, UK

Received by editor(s): March 23, 2000
Published electronically: November 19, 2001
Article copyright: © Copyright 2001 American Mathematical Society