Loewner Theory Research Articles

Loewner Theory for Stochastic Neuron Model

The nonlinear response of the dynamics of the stochastic neuronal activity based on the leaky integrate-and-fire model was reformulated using the Loewner theory. We observed that the signal-to-noise ratio (SNR), which is an indicator of stochastic resonance (SR), and fluctuation-dissipation relation (FDR) for the neuronal dynamics were newly obtained in the theoretical framework of the Loewner evolution. Particularly, we focus on the role of the Loewner entropy [Formula: see text], defined as the entropy of the Loewner driving force, while showing the efficacy of Loewner time conversion to theorize the nonlinear characteristic of the neuronal dynamics. The present results indicate a possible approach to the novel formulation of biophysical models.

Problems related to conformal slit-mappings

In this note we discuss some problems related to conformal slit-mappings. On the one hand, classical Loewner theory leads us to questions concerning the embedding of univalent functions into slit-like Loewner chains. On the other hand, a recent result from monotone probability theory motivates the study of univalent functions from a probabilistic perspective.

Loewner chains with quasiconformal extensions: an approximation approach

A new approach in Loewner Theory proposed by Bracci, Contreras, Díaz-Madrigal and Gumenyuk provides a unified treatment of the radial and the chordal versions of the Loewner equations. In this framework, a generalized Loewner chain satisfies the differential equation $${{\partial {f_t}\left(z \right)} \over {\partial t}} = \left({z - \tau \left(t \right)} \right)\left({1 - \overline {\tau \left(t \right)} z} \right)p\left({z,\;t} \right){{\partial {f_t}\left(z \right)} \over {\partial z}},$$ where τ: [0, ∞) → \(\overline{\mathbb{D}}\) is measurable and p is called a Herglotz function. In this paper, we will show that if there exists a k ⩽ [0, 1) such that p satisfies $$\left| {p\left({z,\;t} \right) - 1} \right| \le k\left| {p\left({z,\;t} \right) + 1} \right|$$ for all z ⩽ \(\mathbb{D}\) and almost all t ⩽ [0, ∞), then, for all t ⩽ [0, ∞), ft has a k-quasiconformal extension to the whole Riemann sphere. The radial case (τ = 0) and the chordal case (τ = 1) have been proven by Becker [J. Reine Angew. Math., vol. 255 (1972), 23–43] and Gumenyuk and the author [Math. Z., vol. 285 (2017), 1063–1089]. In our theorem, no superfluous assumption is imposed on τ ⩽ \(\overline{\mathbb{D}}\). As a key foundation of the proof is an approximation method using a continuous dependence of evolution families and Loewner chains.

The Extension Operators on Bn+1 and Bounded Complete Reinhardt Domains

In this article, we extend the well-known Roper-Suffridge operator on Bn+1 and bounded complete Reinhardt domains in ℂn+1, then we investigate the properties of the generalized operators. Applying the Loewner theory, we obtain the mappings constructed by the generalized operators that have parametric representation on Bn+1. In addition, by using the geometric characteristics and the parametric representation of subclasses of spirallike mappings, we conclude that the extended operators preserve the geometric properties of several subclasses of spirallike mappings on Bn+1 and bounded complete Reinhardt domains in ℂn+1. The conclusions provide new approaches to construct mappings with special geometric properties in ℂn+1.

Approximation and Loewner theory of holomorphic covering mappings

We give conditions in order to approximate locally uniformly holomorphic covering mappings of the unit ball of $${{\mathbb {C}}^{n}}$$ with respect to an arbitrary norm, with entire holomorphic covering mappings. The results rely on a generalization of the Loewner theory for covering mappings which we develop in the paper.

Loewner Theory for Quasiconformal Extensions: Old and New

This survey article gives an account of quasiconformal extensions of univalent functions with its motivational background from Teichmuller theory and classical and modern approaches based on Loewner theory.

Almost spirallike mappings of type and complex order on the unit ball

ABSTRACTFirstly, we give a characterization of almost spirallike mappings of type and complex order on the unit ball in in terms of Loewner chains. Secondly, the growth and covering theorems for almost spirallike mappings of type and complex order on the unit ball are established by Loewner theory. In particular, some well-known results can be obtained using the conclusions in this article.

Quasiconformal extendability of integral transforms of Noshiro-Warschawski functions

Since the nonlinear integral transforms $$ J_{\alpha }[f](z) = \int _{0}^{z}(f'(u))^{\alpha } du $$ and $$ \ \ I_{\alpha }[f](z) =\int _0^z (f(u)/u)^{\alpha } du $$ with a complex number $\alpha $ were introduced, a great number of studies have been dedicated to deriving sufficient conditions for univalence on the unit disk. However, little is known about the conditions where $J_{\alpha }[f]$ or $I_{\alpha }[f]$ produce a holomorphic univalent function in the unit disk which extends to a quasiconformal map on the complex plane. In this paper, we discuss quasiconformal extendability of the integral transforms $J_{\alpha }[f]$ and $I_{\alpha }[f]$ for holomorphic functions which satisfy the Noshiro-Warschawski criterion. Various approaches using pre-Schwarzian derivatives, differential subordination and Loewner theory are applied to this problem.

A Description Method in the Value Region Problem

Fedorov completely described the value region \(\{f(z_0)\}\), \(|z_0|<1\), over the class \(S_R\) of holomorphic univalent functions \(f(z)=z+a_2z^2+\cdots \) in the unit disk \({\mathbb {D}}\) with real coefficients \(a_n\), \(n\ge 2\). We extend this result to the class \(S_R(M)\) of bounded functions \(f\in S_R\) satisfying \(|f(z)|<M\) in \({\mathbb {D}}\). The problem is formulated as the reachable set problem for the Hamilton system of controllable differential equations in the frames of the Loewner theory. A family of Cauchy problems is substituted for the family of boundary value problems. The free parameter in the initial data serves as a parameter for the boundary curve of the value region. The algorithm proposed can be useful in other similar value region problems.

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.

Asymptotically Spirallike Mappings in Reflexive Complex Banach Spaces

In this paper we consider the notion of asymptotic spirallikeness in reflexive complex Banach spaces \(X\), and the connection with univalent subordination chains. Poreda initially introduced the notion of asymptotic starlikeness to characterize biholomorphic mappings on the unit polydisc in \(\mathbb{C }^{n}\) which have parametric representation in the sense of Loewner theory. The authors introduced the notions of \(A\)-asymptotic spirallikeness and \(A\)-parametric representation on the Euclidean unit ball of \(\mathbb{C }^{n}\), where \(A\in L(\mathbb{C }^{n})\) with \(m(A)>0\). They showed that these notions are equivalent whenever \(k_+(A)<2m(A)\). In this paper we prove that if \(k_+(A)<2m(A)\) and \(f\in S(B)\) has \(A\)-parametric representation, then \(f\) is also \(A\)-asymptotically spirallike on the unit ball \(B\) of \(X\). For the converse, we need the additional assumption that \(f\) is a smooth \(A\)-asymptotically spirallike mapping, except in the finite-dimensional case \(X=\mathbb{C }^{n}\) with an arbitrary norm. The notion of asymptotic spirallikeness involves differential equations and may be regarded as giving a geometric characterization of certain domains in \(X\). That is one of the motivations for considering this notion in the case of reflexive complex Banach spaces.

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].

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. (J Reine Angew Math to appear, arXiv:0807.1594; Math Ann 344:947–962, 2009) and Contreras et al. (Revista Matematica Iberoamericana 26:975–1012, 2010) 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 Contreras et al. (Trans Amer Math Soc to appear, 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. Furthermore, 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 Contreras et al. (Trans Amer Math Soc to appear, arXiv:1011.4253) to the general case.

Semigroups versus evolution families in the loewner theory

We show that an evolution family of the unit disc is commuting if and only if the associated Herglotz vector field has separated variables. This is the case if and only if the evolution family comes from a semigroup of holomorphic self-maps of the disc.

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 [Bracci, F., Contreras, M.D. and Díaz-Madrigal, S.: Evolution families and the Loewner equation I: the unit disk. To appear in J. Reine Angew. Math.] 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 [Bracci, F., Contreras, M.D. and Díaz-Madrigal, S.: Evolution families and the Loewner equation I: the unit disk. To appear in J. Reine Angew. Math.]. 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.

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. 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 0807.1594; Bracci F et al. in Math Ann 344:947–962, 2009; Contreras MD et al. in 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. in Complex Analysis. PWN, Warsaw, pp 7–32, 1979; Bauer RO in J Math Anal Appl 302: 484–501, 2005; Goryainov VV and Ba I in Ukrainian Math J 44:1209–1217, 1992).

Parametric representation and asymptotic starlikeness in ℂⁿ

In this paper we consider the notion of asymptotic starlikeness in the Euclidean space C n \mathbb {C}^n . In the case of the maximum norm, asymptotic starlikeness was introduced by Poreda. We have modified his definition slightly, adding a boundedness condition. We prove that the notion of parametric representation which arises in Loewner theory can be characterized in terms of asymptotic starlikeness; i.e. they are equivalent notions. (A regularity assumption of Poreda is not needed.) In particular, starlike mappings and spirallike mappings of type α ∈ ( − π / 2 , π / 2 ) \alpha \in (-\pi /2,\pi /2) are asymptotically starlike. Therefore this notion is a natural generalization of starlikeness. However, we give an example of a spirallike mapping with respect to a linear operator which is not asymptotically starlike. In the case of one complex variable, any function in the class S S is asymptotically starlike; however, in dimension n ≥ 2 n\geq 2 this is no longer true.

A novel class of model constitutive laws in nonlinear elasticity: Construction via Loewner theory

Using a solitonic connection, we show that the class of infinitesimal Backlund transformations originally introduced by Loewner in 1952 in a gasodynamic context results in physically interesting nonlinear model constitutive laws. We obtain laws previously used to model a variety of hard and soft nonlinear elastic responses. A natural extension of the latter leads to a novel class of model constitutive laws where the stress and strain are given parametrically in terms of elliptic functions. Such models allow a change in the concavity of the stress-strain law. Such behavior can be observed in the compression of polycrystalline materials or in the unloading regimes of superelastic nickel-titanium.

On radial stochastic Loewner evolution in multiply connected domains

We discuss the extension of radial SLE to multiply connected planar domains. First, we extend Loewner's theory of slit mappings to multiply connected domains by establishing the radial Komatu–Loewner equation, and show that a simple curve from the boundary to the bulk is encoded by a motion on moduli space and a motion on the boundary of the domain. Then, we show that the vector-field describing the motion of the moduli is Lipschitz. We explain why this implies that “consistent,” conformally invariant random simple curves are described by multidimensional diffusions, where one component is a motion on the boundary, and the other component is a motion on moduli space. We argue what the exact form of this diffusion is (up to a single real parameter κ) in order to model boundaries of percolation clusters. Finally, we show that this moduli diffusion leads to random non-self-crossing curves satisfying the locality property if and only if κ = 6 .

Some Extensions of Loewner's Theory of Monotone Operator Functions

Several extensions of Loewner's theory of monotone operator functions are given. These include a theorem on boundary interpolation for matrix-valued functions in the generalized Nevanlinna class. The theory of monotone operator functions is generalized from scalar-to matrix-valued functions of an operator argument. A notion of κ-monotonicity is introduced and characterized in terms of classical Nevanlinna functions with removable singularities on a real interval. Corresponding results for Stieltjes functions are presented.