On the geometry of the Teichmueller space.
Item
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Title
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On the geometry of the Teichmueller space.
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Identifier
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AAI9530894
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identifier
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9530894
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Creator
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Lakic, Nikola.
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Contributor
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Adviser: Frederick P. Gardiner
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Date
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1995
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Language
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English
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Publisher
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City University of New York.
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Subject
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Mathematics
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Abstract
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Let A(X) be the Banach space of integrable, holomorphic, quadratic differentials {dollar}\varphi{dollar} on a Riemann surface X. We characterize the points of A(X) at which the norm is weak uniformly convex in terms of the infinitesimal form of Teichmuller's metric on QS mod S and we give a quantified version of this characterization. Sullivan's coiling property applies along any Beltrami line (t{dollar}\vert\varphi\vert/\varphi{dollar}) for which {dollar}\varphi{dollar} is a point of weak uniform convexity and the amount of coiling is quantified by the quantified version of weak convexity. For a closed set J in {dollar}\doubc{dollar}, we let A(J) be the Banach space of integrable functions in {dollar}\doubc{dollar} which are holomorphic in the complement of J. We generalize Bers' approximation theorem by showing that rational functions with simple poles in J are dense in A(J). Density is with respect to the {dollar}L\sp1{dollar}-norm over the whole complex plane, including J. Assume both X and Y are Riemann surfaces which are subsets of compact Riemann surfaces {dollar}X\sb1{dollar} and {dollar}Y\sb1{dollar}, respectively, such that set {dollar}X\sb1{dollar}-X has infinitely many points. Finally we prove that the only subjective complex linear isometries between spaces of integrable holomorphic quadratic differentials on X and Y are the ones induced by conformal homeomorphisms and complex constants of modulus 1. As a corollary we conclude that if the Teichmuller space of X is biholomorphic to the Teichmuller space of Y, then X is quasiconformal to Y.
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Type
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dissertation
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Source
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PQT Legacy CUNY.xlsx
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degree
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Ph.D.
