Pavel Gumenyuk

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.

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 [10, 11, 16] 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 nonautonomous Berkson–Porta representation of Herglotz vector fields in the unit disk [10].

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

We study boundary singularities which can appear for infinitesimal generators of one-parameter semigroups of holomorphic self-maps of the unit disc. We introduce “regular” fractional singularities and characterize them in terms of the behavior of the associated semigroups and Kœnigs functions. We also provide necessary and sufficient geometric criteria on the shape of the image of the Kœnigs 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.

Chordal Loewner chains with quasiconformal extensions

In 1972, Becker (J Reine Angew Math 255:23–43, 1972), 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 (J Reine Angew Math 354:74–94, 1984) for functions in the half-plane.

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

We introduce a notion of hyperbolicity and parabolicity for a holomorphic self-map

Quasiconformal extensions, Loewner chains, and the \(\lambda \)-Lemma

f: \Delta^N \to \Delta^N

Parametric Representations and Boundary Fixed Points of Univalent Self-Maps of the Unit Disk

of the polydisc which does not admit fixed points in

Caratheodory convergence of immediate basins of attraction to a Siegel disk

\Delta^N

Local duality in Loewner equations

. We generalize to the polydisc two classical one-variable results: we solve the Valiron equation for a hyperbolic

Geometry Behind Chordal Loewner Chains

f