Loewner theory in annulus I: Evolution families and differential equations
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- by Manuel D. Contreras, Santiago Díaz-Madrigal and Pavel Gumenyuk PDF
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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.References
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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
- MR Author ID: 335888
- Email: contreras@us.es
- 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
- MR Author ID: 310764
- Email: madrigal@us.es
- 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
- MR Author ID: 706440
- Email: Pavel.Gumenyuk@math.uib.no, gumenyuk@axp.mat.uniroma2.it
- 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 - © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 365 (2013), 2505-2543
- MSC (2010): Primary 30C35, 30C20, 30D05; Secondary 30C80, 34M15
- DOI: https://doi.org/10.1090/S0002-9947-2012-05718-7
- MathSciNet review: 3020107