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Pavel Gumenyuk

Publications and source records attributed to Pavel Gumenyuk.

35 records · Page 2Linked to original sources
arXiv2017

Quasiconformal extensions, Loewner chains, and the lambda-Lemma

In 1972, J. Becker [J. Reine Angew. Math. 255] discovered a sufficient condition for quasiconformal extendibility of Loewner chains. Many known conditions for quasiconformal extendibility of holomorphic functions in the unit disk can be deduced from his result. We give a new proof of (a generalization of) Becker's result based on Slodkowski's Extended lambda-Lemma. Moreover, we characterize all quasiconformal extensions produced by Becker's (classical) construction and use that to obtain examples in which Becker's extension is extremal (i.e. optimal in the sense of maximal dilatation) or, on the contrary, fails to be extremal.

math.CV↗
arXiv2017

Parametric representation of univalent functions with boundary regular fixed points

Given a set $\mathfrak S$ of conformal maps of the unit disk $\mathbb D$ into itself that is closed under composition, we address the question whether $\mathfrak S$ can be represented as the reachable set of a Loewner - Kufarev - type ODE $\mathrm{d}w_t/\mathrm{d}t=G_t\circ w_t$, $w_0=\mathsf{id}_{\mathbb D}$, where the control functions $t\mapsto G_t$ form a convex cone $\mathcal M_{\mathfrak S}$. For the set of all conformal $φ:\mathbb D\to\mathbb D$ with $φ(0)=0$, $φ'(0)>0$, an affirmative answer to this question is the essence of Loewner's well-known Parametric Representation of univalent functions [Math. Ann. 89 (1923), 103-121]. In this paper, we study classes of conformal self-maps defined by their boundary regular fixed points and, in part of the cases, establish their Loewner-type representability.

math.CV↗
arXiv2017

Value regions of univalent self-maps with two boundary fixed points

In this paper we find the exact value region $\mathcal V(z_0,T)$ of the point evaluation functional $f\mapsto f(z_0)$ over the class of all holomorphic injective self-maps $f:\mathbb D\to\mathbb D$ of the unit disk $\mathbb D$ having a boundary regular fixed point at $σ=-1$ with $f'(-1)=e^{T}$ and the Denjoy - Wolff point at $τ=1$.

math.CV↗
arXiv2017

Parametric representations and boundary fixed points of univalent self-maps of the unit disk

A classical result in the theory of Loewner's parametric representation states that the semigroup $\mathfrak U_*$ of all conformal self-maps $ϕ$ of the unit disk $\mathbb{D}$ normalized by $ϕ(0) = 0$ and $ϕ'(0) > 0$ can be obtained as the reachable set of the Loewner - Kufarev control system $$ \frac{\mathrm{d} w_t}{\mathrm{d} t}=G_t\circ w_t,\quad t\geqslant0,\qquad w_0=\mathsf{id}_{\mathbb{D}}, $$ where the control functions $t\mapsto G_t\in\mathsf{Hol}(\mathbb{D},\mathbb{C})$ form a certain convex cone. Here we extend this result to the semigroup $\mathfrak U[F]$ consisting of all conformal $ϕ:\mathbb{D}\to\mathbb{D}$ whose set of boundary regular fixed points contains a given finite set $F\subset\partial\mathbb{D}$ and to its subsemigroup $\mathfrak U_τ[F]$ formed by $\mathsf{id}_{\mathbb{D}}$ and all $ϕ\in\mathfrak U[F]\setminus\{\mathsf{id}_{\mathbb{D}}\}$ with the prescribed boundary Denjoy - Wolff point $τ\in\partial\mathbb{D}\setminus F$. This completes the study launched in [P. Gumenyuk, Preprint 2016, ArXiv:1603.04043], where the case of interior Denjoy - Wolff point $τ\in\mathbb{D}$ was considered.

math.CV↗
arXiv2016

Valiron and Abel equations for holomorphic self-maps of the polydisc

We introduce a notion of hyperbolicity and parabolicity for a holomorphic self-map $f: Δ^N \to Δ^N$ of the polydisc which does not admit fixed points in $Δ^N$. We generalize to the polydisc two classical one-variable results: we solve the Valiron equation for a hyperbolic $f$ and the Abel equation for a parabolic nonzero-step $f$. This is done by studying the canonical Kobayashi hyperbolic semi-model of $f$ and by obtaining a normal form for the automorphisms of the polydisc. In the case of the Valiron equation we also describe the space of all solutions.

math.CV↗
arXiv2015

Chordal Loewner chains with quasiconformal extensions

In 1972, Becker [J. Reine Angew. Math. 255 (1972), 23-43] discovered a construction of quasiconformal extensions making use of the classical radial Loewner chains. In this paper we develop a chordal analogue of Becker's construction. As an application, we establish new sufficient conditions for quasiconformal extendibility of holomorphic functions and give a simplified proof of one well-known result by Becker and Pommerenke for functions in the half-plane [J. Reine Angew. Math. 354 (1984), 74-94].

math.CV↗
arXiv2013

Contact points and fractional singularities for semigroups of holomorphic self-maps in the unit disc

We study boundary singularities which can appear for infinitesimal generators of one-parameter semigroups of holomorphic self-maps in the unit disc. We introduce "regular" fractional singularities and characterize them in terms of the behavior of the associated semigroups and Koenigs functions. We also provide necessary and sufficient geometric criteria on the shape of the image of the Koenigs function for having such singularities. In order to do this, we study contact points of semigroups and prove that any contact (not fixed) point of a one-parameter semigroup corresponds to a maximal arc on the boundary to which the associated infinitesimal generator extends holomorphically as a vector field tangent to this arc.

math.CV↗
arXiv2013

Chordal Loewner Equation

The aim of this survey paper is to present a complete direct proof of the well celebrated cornerstone result in Loewner Theory, originally due to Kufarev et al [Trudy Tomsk. Gos. Univ. Ser. Meh.-Mat. 200 (1968) 142-164. MR0257336 (41 #1987)], stating that the family of the hydrodynamically normalized conformal self-maps of the upper-half plane onto the complement of a gradually erased slit satisfies, under a suitable parametrization, the chordal Loewner differential equation. The proof is based solely on basic theorems of Geometric Function Theory combined with some elementary topological facts and does not require any advanced technique.

math.CV↗
arXiv2013

Boundary regular fixed points in Loewner theory

We characterize regular fixed points of evolution families in terms of analytical properties of the associated Herglotz vector fields and geometrical properties of the associated Loewner chains. We present several examples showing the rôle of the given conditions. Moreover, we study the relations between evolution families and Herglotz vector fields at regular contact points and prove an embedding result for univalent self-maps of the unit disc with a given boundary regular fixed point into an evolution family with prescribed boundary data.

math.CV↗
arXiv2013

Angular and unrestricted limits of one-parameter semigroups in the unit disk

We study local boundary behaviour of one-parameter semigroups of holomorphic functions in the unit disk. Earlier under some addition condition (the position of the Denjoy - Wolff point) it was shown in [M.D.Contreras, S.Diaz-Madrigal and Ch.Pommerenke, Ann. Acad. Sci. Fenn. Math. 29(2004), No.2, 471-488] that elements of one-parameter semigroups have angular limits everywhere on the unit circle and unrestricted limits at all boundary fixed points. We prove stronger versions of these statements with no assumption on the position of the Denjoy - Wolff point. In contrast to many other problems, in the question of existence for unrestricted limits it appears to be more complicated to deal with the boundary Denjoy - Wolff point (the case not covered in [M.D.Contreras, S.Diaz-Madrigal and Ch.Pommerenke, Ann. Acad. Sci. Fenn. Math. 29(2004), No.2, 471-488]) than with all the other boundary fixed points of the semigroup.

math.CV↗
arXiv2012

Local duality in Loewner equations

Among diversity of frameworks and constructions introduced in Loewner Theory by different authors, one can distinguish two closely related but still different ways of reasoning, which colloquially may be described as "increasing" and "decreasing". In this paper we review in short the main types of (deterministic) Loewner evolution discussed in the literature and describe in detail the local duality between "increasing" and "decreasing" cases within the general unifying approach in Loewner Theory proposed recently in [Bracci et al. to appear in J Reine Angew Math; arXiv:0807.1594v1], [Bracci et al. in Math Ann 344:947-962, 2009; arXiv:0807.1715v1], [Contreras et al. in Revista Matemática Iberoamericana 26:975-1012, 2010; arXiv:0902.3116v1]. In particular, we extend several results of R.O.Bauer [in J Math Anal Appl 302:484-501, 2005; arXiv:math/0306130v1], which deal with the chordal Loewner evolution, to this general setting. Although the duality is given by a simple change of the parameter, not all the results for the "decreasing" case can be obtained by mere translating the corresponding results for the "increasing" case. In particular, as a byproduct of establishing local duality between evolution families and their "decreasing" counterparts we obtain a new characterization of generalized Loewner chains.

math.CV↗
arXiv2011

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 [Bracci et al, to appear in J. Reine Angew. Math. Available on ArXiv 0807.1594; Bracci et al, Math. Ann. 344(2009), 947--962; Contreras et al, Revista Matematica Iberoamericana 26(2010), 975--1012] 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 [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. Futhermore, 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 [ArXiv 1011.4253] to the general case.

math.CV↗
arXiv2010

Loewner Theory in annulus I: evolution families and differential equations

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 famous Bieberbach's conjecture and well-celebrated Schramm's Stochastic Loewner Evolution (SLE). Recently Bracci et al [Bracci et al, to appear in J. Reine Angew. Math. Available on ArXiv 0807.1594; Bracci et al, Math. Ann. 344(2009), 947--962; Contreras et al, Revista Matematica Iberoamericana 26(2010), 975--1012] have proposed a new approach bringing together all the variants of the (deterministic) Loewner Evolution in a simply connected reference domain. We construct an analogue 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 1-to-1 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 [Bracci et al, to appear in J. Reine Angew. Math. Available on ArXiv 0807.1594].

math.CV↗
arXiv2010

Geometry behind chordal Loewner chains

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 (SLE). It provides analytic description of expanding domains dynamics in the plane. Two cases have been developed in the classical theory, namely the {\it radial} and the {\it 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 [arXiv:0807.1594v1, arXiv:0807.1715v1, arXiv:0902.3116v1] 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 by I.A.Aleksandrov, S.T.Aleksandrov and V.V.Sobolev, 1983; V.V.Goryainov and I.Ba, 1992; R.O.Bauer, 2005 [arXiv:math/0306130v1].

math.CV↗
arXiv2009

Loewner chains in the unit disk

In this paper we introduce a general version of the notion of Loewner chains which comes from the new and unified treatment, given in [arXiv:0807.1594], of the radial and chordal variant of the Loewner differential equation, which is of special interest in geometric function theory as well as for various developments it has given rise to, including the famous Schramm-Loewner evolution. In this very general setting, we establish a deep correspondence between these chains and the evolution families introduced in [arXiv:0807.1594]. Among other things, we show that, up to a Riemann map, such a correspondence is one-to-one. In a similar way as in the classical Loewner theory, we also prove that these chains are solutions of a certain partial differential equation which resembles (and includes as a very particular case) the classical Loewner-Kufarev PDE.

math.CV↗
arXiv2008

Matching univalent functions and conformal welding

Given a conformal mapping $f$ of the unit disk $\mathbb D$ onto a simply connected domain $D$ in the complex plane bounded by a closed Jordan curve, we consider the problem of constructing a matching conformal mapping, i.e., the mapping of the exterior of the unit disk $\mathbb D^*$ onto the exterior domain $D^*$ regarding to $D$. The answer is expressed in terms of a linear differential equation with a driving term given as the kernel of an operator dependent on the original mapping $f$. Examples are provided. This study is related to the problem of conformal welding and to representation of the Virasoro algebra in the space of univalent functions.

math.CV↗