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Transactions of the American Mathematical Society

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Loewner theory in annulus I: Evolution families and differential equations

Authors: Manuel D. Contreras, Santiago Díaz-Madrigal and Pavel Gumenyuk
Journal: Trans. Amer. Math. Soc. 365 (2013), 2505-2543
MSC (2010): Primary 30C35, 30C20, 30D05; Secondary 30C80, 34M15
Published electronically: November 1, 2012
MathSciNet review: 3020107
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Abstract: Loewner theory, based on dynamical viewpoint, is a powerful tool in complex analysis, which plays a crucial role in such important achievements as the proof of the famous Bieberbach conjecture and the well-celebrated Schramm stochastic Loewner evolution (SLE). Recently, Bracci et al. proposed a new approach bringing together all the variants of the (deterministic) Loewner evolution in a simply connected reference domain. We construct an analog of this theory for the annulus. In this paper, the first of two articles, we introduce a general notion of an evolution family over a system of annuli and prove that there is a one-to-one correspondence between such families and semicomplete weak holomorphic vector fields. Moreover, in the non-degenerate case, we establish a constructive characterization of these vector fields analogous to the non-autonomous Berkson-Porta representation of Herglotz vector fields in the unit disk.

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Additional Information

Manuel D. Contreras
Affiliation: Camino de los Descubrimientos, s/n, Departamento de Matemática Aplicada II, Escuela Técnica Superior de Ingeniería, Universidad de Sevilla, Sevilla, 41092, Spain

Santiago Díaz-Madrigal
Affiliation: Camino de los Descubrimientos, s/n, Departamento de Matemática Aplicada II, Escuela Técnica Superior de Ingeniería, Universidad de Sevilla, Sevilla, 41092, Spain

Pavel Gumenyuk
Affiliation: Department of Mathematics, University of Bergen, Johannes Brunsgate 12, Bergen 5008, Norway
Address at time of publication: Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 1. 00133 Roma, Italy

Keywords: Univalent functions, annulus, Loewner chains, Loewner evolution, evolution family, parametric representation
Received by editor(s): November 18, 2010
Received by editor(s) in revised form: September 8, 2011
Published electronically: November 1, 2012
Additional Notes: The first and second authors were partially supported by the Ministerio de Ciencia e Innovación and the European Union (FEDER), project MTM2009-14694-C02-02
The authors were partially supported by the ESF Networking Programme “Harmonic and Complex Analysis and its Applications” and by La Consejería de Economía, Innovación y Ciencia de la Junta de Andalucía (research group FQM-133)
The third author was supported by a grant from Iceland, Liechtenstein, and Norway through the EEA Financial Mechanism. Supported and coordinated by Universidad Complutense de Madrid and by Instituto de Matemáticas de la Universidad de Sevilla. Partially supported by the Scandinavian Network “Analysis and Applications” (NordForsk), project #080151, and the Research Council of Norway, project #177355/V30
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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