Loewner chains and parametric representation in several complex variables |
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Authors: | Ian Graham Gabriela Kohr |
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Institution: | a Department of Mathematics, University of Toronto, Toronto, ON M5S 3G3, Canada b Faculty of Mathematics and Computer Science, Babe?-Bolyai University, 1 M. Kog?lniceanu Str., 3400 Cluj-Napoca, Romania |
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Abstract: | Let B be the unit ball of with respect to an arbitrary norm. We study certain properties of Loewner chains and their transition mappings on the unit ball B. We show that any Loewner chain f(z,t) and the transition mapping v(z,s,t) associated to f(z,t) satisfy locally Lipschitz conditions in t locally uniformly with respect to z∈B. Moreover, we prove that a mapping f∈H(B) has parametric representation if and only if there exists a Loewner chain f(z,t) such that the family {e−tf(z,t)}t?0 is a normal family on B and f(z)=f(z,0) for z∈B. Also we show that univalent solutions f(z,t) of the generalized Loewner differential equation in higher dimensions are unique when {e−tf(z,t)}t?0 is a normal family on B. Finally we show that the set S0(B) of mappings which have parametric representation on B is compact. |
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Keywords: | Loewner chain Transition mapping Parametric representation Univalent mapping Loewner differential equation |
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