Semigroup of conformal mappings of the upper half-plane into itself with hydrodynamic normalization at infinity

Goryainov, V. V.; Bâ, Idrissa

doi:10.1007/bf01057676

Cited by 33 publications

(47 citation statements)

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“…5 we state and proof a result (Theorem 5.5), which is an analogue of Theorem 1.10 for a subclass of chordal evolution families having some additional regularity, which were considered in [26]. Theorems 1.10 and 5.5 give necessary and sufficient conditions for an inclusion chain to be the range of a Loewner chain of given type.…”

Section: Problemmentioning

confidence: 95%

Geometry Behind Chordal Loewner Chains

Contreras

Díaz-Madrigal

Gumenyuk

2010

Complex Anal. Oper. Theory

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Loewner Theory is a deep technique in Complex Analysis affording a basis for many further important developments such as the proof of famous Bieberbach's conjecture and well-celebrated Schramm's stochastic Loewner evolution. It provides analytic description of expanding domains dynamics in the plane. Two cases have been developed in the classical theory, namely the radial and the chordal Loewner evolutions, referring to the associated families of holomorphic self-mappings being normalized at an internal or boundary point of the reference domain, respectively. Recently there has been introduced a new approach (Bracci F et al. in Evolution families and the Loewner equation I: the unit disk. Preprint 2008. Available on ArXiv Communicated by Dr. Alexander Vasiliev. M. D. 542 M. D. Contreras et al. 0807.1594; Bracci F et al. in Math Ann 344:947-962, 2009; Loewner chains in the unit disk. To appear in Revista Matemática Iberoamericana; preprint available at arXiv:0902.3116v1 [math.CV]) bringing together, and containing as quite special cases, radial and chordal variants of Loewner Theory. In the framework of this approach we address the question what kind of systems of simply connected domains can be described by means of Loewner chains of chordal type. As an answer to this question we establish a necessary and sufficient condition for a set of simply connected domains to be the range of a generalized Loewner chain of chordal type. We also provide an easy-to-check geometric sufficient condition for that. In addition, we obtain analogous results for the less general case of chordal Loewner evolution considered in (Aleksandrov IA et al.

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Section: Problemmentioning

confidence: 95%

Geometry Behind Chordal Loewner Chains

Contreras

Díaz-Madrigal

Gumenyuk

2010

Complex Anal. Oper. Theory

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“…We cite the pioneering works of Goryainov [12] and Goryainov and Ba [13]. After that, Schramm [25] and Lawler, Schramm and Werner [15], [16] proved the Mandelbroit conjecture using a stochastic version of this chordal Loewner equation.…”

Section: 1)mentioning

confidence: 99%

“…As we have remarked in the introduction, this case is much more complicated and apart from a couple of papers due to Goryainov and Ba [12] , [13], there are no references till the end of the nineties where a series of paper of G.F. Lawler, O. Schramm, W. Werner and Bauer appeared [25], [15], [16], [2].…”

Section: Evolution Families With a Common Boundary Fixed Pointmentioning

confidence: 99%

Evolution families and the Loewner equation I: the unit disc

Bracci

Contreras

Díaz-Madrigal

2012

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. In this paper we introduce a general version of the Loewner differential equation which allows us to present a new and unified treatment of both the radial equation introduced in 1923 by K. Loewner and the chordal equation introduced in 2000 by O. Schramm. In particular, we prove that evolution families in the unit disc are in one to one correspondence with solutions to this new type of Loewner equations. Also, we give a Berkson-Porta type formula for non-autonomous weak holomorphic vector fields which generate such Loewner differential equations and study in detail geometric and dynamical properties of evolution families.

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“…The first equality in (2) follows from the well-known Nevanlinna representation, cf. [13,Theorem 5.3] or [7], via the Stieltjes inversion formula. The second equality is due to the decomposition…”

Section: Lemma 24 Let T T ∈ [0 1] and T ≤ T Then (1) The Functmentioning

confidence: 99%

“…The authors would like to thank the anonymous referee for many valuable comments and a careful reading of an earlier version of this manuscript. We are especially grateful for the bringing to our attention of the references [1,7]. The second author gratefully acknowledges the encouragement of Prof. Dr. Robert Tichy.…”

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confidence: 99%

A Loewner Equation for Infinitely Many Slits

Technau

2016

Comput. Methods Funct. Theory

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It is well-known that the growth of a slit in the upper half-plane can be encoded via the chordal Loewner equation, which is a differential equation for schlicht functions with a certain normalisation. We prove that a multiple slit Loewner equation can be used to encode the growth of the union of multiple slits in the upper half-plane if the slits have pairwise disjoint closures. Under certain assumptions on the geometry of , our approach allows us to derive a Loewner equation for infinitely many slits as well.

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Semigroup of conformal mappings of the upper half-plane into itself with hydrodynamic normalization at infinity

Cited by 33 publications

References 1 publication

Geometry Behind Chordal Loewner Chains

Geometry Behind Chordal Loewner Chains

Evolution families and the Loewner equation I: the unit disc

A Loewner Equation for Infinitely Many Slits

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