PUBLICATIONS DE L'INSTITUT MATHEMATIQUE (BEOGRAD) (N.S.)Vol. 75(89), pp. 199–215 (2004)
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PUBLICATIONS DE L'INSTITUT MATHEMATIQUE (BEOGRAD) (N.S.) Vol. 75(89), pp. 199–215 (2004) |
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LOEWNER CHAINS AND BIHOLOMORPHIC MAPPINGS IN $\mathbb{C}^n$ AND REFLEXIVE COMPLEX BANACH SPACESIan Graham, Gabriela Kohr, and John A. PfaltzgraffDepartment of Mathematics, University of Toronto, Toronto, Ontario M5S 3G3, Canada and Faculty of Mathematics and Computer Science, Babes-Bolyai University, 1 M. Kogalniceanu Str., 3400 Cluj-Napoca, Romania and Department of Mathematics, CB 3250, University of North Carolina, Chapel Hill, NC 27599-3250, USAAbstract: This paper is a survey of very recent results about biholomorphic mappings of the ball in $\Bbb{C}^n$ and in reflexive complex Banach spaces. After recalling existence and regularity results in $\Bbb{C}^n$, we present certain applications including univalence criteria and quasiconformal extension results. We also consider nonuniqueness phenomena for solutions of the Loewner differential equation, and a geometric characterization of Loewner chains which satisfy a growth condition in $t$ based on a generalization of the Carathéodory convergence theorem. Finally we describe some properties of Loewner chains and the Loewner equation on the unit ball of a reflexive complex Banach space. Keywords: Carathéodory class, Loewner chain, Loewner differential equation, transition mapping, kernel convergence, starlike mapping, convex mapping, close-to-starlike mapping Classification (MSC2000): 32H02; 30C45 Full text of the article: (for faster download, first choose a mirror)
Electronic fulltext finalized on: 27 Oct 2004. This page was last modified: 22 Feb 2005.
© 2004 Mathematical Institute of the Serbian Academy of Science and Arts
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