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Loewner Theory in annulus II: Loewner chains

Loewner Theory in annulus II: Loewner chains Loewner Theory, based on dynamical viewpoint, proved itself to be a powerful tool in Complex Analysis and its applications. Recently Bracci et al. (J Reine Angew Math to appear, arXiv:0807.1594; Math Ann 344:947–962, 2009) and Contreras et al. (Revista Matemática Iberoamericana 26:975–1012, 2010) have proposed a new approach bringing together all the variants of the (deterministic) Loewner Evolution in a simply connected reference domain. This paper is devoted to the construction of a general version of Loewner Theory for the annulus launched in Contreras et al. (Trans Amer Math Soc to appear, arXiv:1011.4253). We introduce the general notion of a Loewner chain over a system of annuli and obtain a 1-to-1 correspondence between Loewner chains and evolution families in the doubly connected setting similar to that in the Loewner Theory for the unit disk. Furthermore, we establish a conformal classification of Loewner chains via the corresponding evolution families and via semicomplete weak holomorphic vector fields. Finally, we extend the explicit characterization of the semicomplete weak holomorphic vector fields obtained in Contreras et al. (Trans Amer Math Soc to appear, arXiv:1011.4253) to the general case. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

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Publisher
Springer Journals
Copyright
Copyright © 2011 by Springer Basel AG
Subject
Mathematics; Mathematical Methods in Physics; Analysis
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-011-0016-z
Publisher site
See Article on Publisher Site

Abstract

Loewner Theory, based on dynamical viewpoint, proved itself to be a powerful tool in Complex Analysis and its applications. Recently Bracci et al. (J Reine Angew Math to appear, arXiv:0807.1594; Math Ann 344:947–962, 2009) and Contreras et al. (Revista Matemática Iberoamericana 26:975–1012, 2010) have proposed a new approach bringing together all the variants of the (deterministic) Loewner Evolution in a simply connected reference domain. This paper is devoted to the construction of a general version of Loewner Theory for the annulus launched in Contreras et al. (Trans Amer Math Soc to appear, arXiv:1011.4253). We introduce the general notion of a Loewner chain over a system of annuli and obtain a 1-to-1 correspondence between Loewner chains and evolution families in the doubly connected setting similar to that in the Loewner Theory for the unit disk. Furthermore, we establish a conformal classification of Loewner chains via the corresponding evolution families and via semicomplete weak holomorphic vector fields. Finally, we extend the explicit characterization of the semicomplete weak holomorphic vector fields obtained in Contreras et al. (Trans Amer Math Soc to appear, arXiv:1011.4253) to the general case.

Journal

Analysis and Mathematical PhysicsSpringer Journals

Published: Nov 29, 2011

References