We present a new geometric construction of Loewner chains in one and several complex variables which holds on a complete hyperbolic complex manifold M and prove that there is essentially a one-to-one correspondence between evolution families of order d and Loewner chains of the same order. As a consequence we obtain a solution for any Loewner-Kufarev PDE, given by univalent mappings (f_t) from M to a complex manifold N. The problem of finding solutions given by univalent mappings with range in C^n is reduced to investigating whether the union of the images f_t(M) is biholomorphic to a domain in C^n. We apply such results to the study of univalent mappings from the unit ball B^n to C^n.
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 cor…
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 [A…
In paper found conditions that guarantee that solution of Loewner-Kufarev equation maps unit disc onto domain with quasiconformal rectifiable boundary, or it has continuation on closed unit disc, or it's inverse function has continuation on closure of domain.
We discuss the problem of embedding univalent functions into Loewner chains in higher dimension. In particular, we prove that a normalized univalent map of the ball in $\C^n$ whose image is a smooth strongly pseudoconvex domain is embeddable into a normalized Loewner chain (satisfying also some extra regularity properties) if and only if the closure of the image is polynomially convex.
In this article, we obtain quasiconformal extensions of some classes of conformal maps defined either on the unit disc or on the exterior of it onto the extended complex plane. Some of these extensions have been obtained by constructing suitable Loewner chains and others have been obtained by applying a well-known result.
We find a solution to the Loewner chain equation in the case when the infinitesimal generator satisfies h(0,t)=0, Dh(0,t)=A for any linear operator with m(A)>0. We also study the related classes of spirallike mappings, mappings with parametric representation and asymptotically spirallike mappings.
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 h…
To explore the relation between properties of Loewner chains and properties of their driving functions, we study Loewner chains driven by functions $U$ of finite total variation. Under some appropriate conditions, we show existence of the simple trace $γ$ and establish continuity of the map $U$ to $γ$ with respect to uniform topology on $γ$ and the total variation topology on $U$. In the spirit of work of Wong and Tran-Lind, we also obtain conditions on the driving function that ensures the trace to be continuously differentiable.
In this paper we confirm that several crucial theorems due to Pommerenke and Becker for the theory of Loewner chains work well without normalization on the complex-valued first coefficient. As applications of those considerations, some new univalent and quasiconformal extension criteria are given in the last section.
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].
We construct a holomorphic embedding $ϕ:\mathbb B^3\rightarrow\mathbb C^3$ such that $ϕ(\mathbb B^3)$ is not Runge in any strictly larger domain. As a consequence, $\mathcal S\neq\mathcal S^1$ for $n=3$.
The object of the present paper is to obtain a more general condition for univalence of meromorphic functions in the U*. The significant relationships and relevance with other results are also given. A number of known univalent conditions would follow upon specializing the parameters involved in our main results.
In this paper we study the class of "shearing" holomorphic maps of the unit ball of the form $(z,w)\mapsto (z+g(w), w)$. Besides general properties, we use such maps to construct an example of a normalized univalent map of the ball onto a Runge domain in $\mathbb C^n$ which however cannot be embedded into a Loewner chain whose range is $\mathbb C^n$.
We introduce a family of natural normalized Loewner chains in the unit ball, which we call "geräumig"---spacious---which allow to construct, by means of suitable variations, other normalized Loewner chains which coincide with the given ones from a certain time on. We apply our construction to the study of support points, extreme points and time-$\log M$-reachable functions in the class $S^0$ of mappings admitting parametric representation.
This paper defines the notion of generators for a class of decreasing radial Loewner chains which are only continuous with respect to time. For this purpose, "Loewner's integral equation" which generalizes Loewner's differential equation is defined and analyzed. The definition of generators is motivated by the Lévy-Khintchine representation for additive processes on the unit circle. Actually, we can and do introduce a homeomorphism between the above class of Loewner chains and the set of the distributions of increments of additive processes equipped with suitable topologies. On the other hand,…
In this paper we obtain, by the method of Loewner chains, some sufficient conditions for the analyticity and the univalence of the functions defined by an integral operator. These conditions in- volves Ruscheweyh and Salagean derivative operator in the open unit disk. In particular cases, we find the well-known conditions for univalency established by Becker [3], Ahlfors [2], Kanas and Srivastava [8] and others for analytic mappings f : U ! C: Also, we obtain the corresponding new, useful and simpler conditions for this integral operator.
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 t…
We consider a univalent analytic function $f$ on the half-plane satisfying the condition that the supremum norm of its (pre-)Schwarzian derivative vanishes on the boundary. Under certain extra assumptions on $f$, we show that there exists a chordal Loewner chain initiated from $f$ until some finite time, and this Loewner chain defines a quasiconformal extension of $f$ over the boundary such that its complex dilatation is given explicitly in terms of the (pre-)Schwarzian derivative in some neighborhood of the boundary. This can be regarded as the half-plane version of the corresponding result d…
We prove one-to-one correspondences between certain decreasing Loewner chains in the upper half-plane, a special class of real-valued Markov processes, and quantum stochastic processes with monotonically independent additive increments. This leads us to a detailed investigation of probability measures on $\mathbb{R}$ with univalent Cauchy transform. We discuss several subclasses of such measures and obtain characterizations in terms of analytic and geometric properties of the corresponding Cauchy transforms. Furthermore, we obtain analogous results for the setting of decreasing Loewner chains …
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 $$ \partial_{t}f_{t}(z) = (z - τ(t))(1-\overline{τ(t)}z)\partial_{z}f_{t}(z)p(z,t), $$ where $τ: [0,\infty) \to \overline{\mathbb{D}}$ is measurable and $p$ is called a Herglotz function. In this paper, we will show that if there exists a $k \in [0,1)$ such that $p$ satisfies $$ |p(z,t) - 1| \leq k |p(z,t) + 1| $$ for all $z \in \…
It is known that a backward Schramm--Loewner evolution (SLE) is coupled with a free boundary Gaussian free field (GFF) with boundary perturbation to give conformal welding of quantum surfaces. Motivated by a generalization of conformal welding for quantum surfaces with multiple marked boundary points, we propose a notion of multiple backward SLE. To this aim, we investigate the commutation relation between two backward Loewner chains, and consequently, we find that the driving process of each backward Loewner chain has to have a drift term given by logarithmic derivative of a partition functio…
We study some features of the energy of a deterministic chordal Loewner chain, which is defined as the Dirichlet energy of its driving function. In particular, using an interpretation of this energy as a large deviation rate function for SLE$_κ$ as $κ$ tends to 0 and the known reversibility of the SLE$_κ$ curves for small $κ$, we show that the energy of a deterministic curve from one boundary point A of a simply connected domain D to another boundary point B, is equal to the energy of its time-reversal ie. of the same curve but viewed as going from B to A in D.
Let R be a hyperbolic Riemann surface with boundary $\partial R$ and suppose that $γ:[0,T]\to R\cup\partial R$ is a simple curve growing from the boundary of R. By lifting $R_{t}=R\setminus γ(0,t]$ to the universal covering space of R (which we assume is the upper half-plane $\mathbb{H}=\{z\in\mathbb{C}:Im[z]>0\}$) via the covering map $π:\mathbb{H}\to R$, we can define a family of simply-connected domains $D_{t}=π^{-1}(R_{t})$. For each $t\in[0,T]$, suppose that f_{t} is a conformal map of \mathbb{H} onto D_{t} such that f(z,t)=f_{t}(z) is differentiable almost everywhere in (0,T) with respec…
The Loewner equation is known as a one-dimensional reduction of the Benney chain as well as the dispersionless KP hierarchy. We propose a reverse process showing that time splitting in the Loewner or the Loewner-Kufarev equation leads to some known integrable systems.
The Loewner equation provides a correspondence between continuous real-valued functions $λ_t$ and certain increasing families of half-plane hulls $K_t$. In this paper we study the deterministic relationship between specific analytic properties of $λ_t$ and geometric properties of $K_t$. Our motivation comes, however, from the stochastic Loewner equation (SLE$_κ$), where the associated function $λ_t$ is a scaled Brownian motion and the corresponding domains $\mathbb{H} \backslash K_t$ are Hölder domains. We prove that if the increasing family $K_t$ is generated by a simple curve and the final d…
We introduce a process, that we call "shearing", which for any given normal Loewner chain produces a normal Loewner chain made of shears automorphisms. As an application, and in stringent contrast to the one-dimensional case, we prove the existence of a starlike bounded function in the class $S^0$ of the ball $\mathbb B^2$ (in fact the restriction of a shear automorphism of $\mathbb C^2$) which is a support point for a linear continuous functional.
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.
By using the method of Loewner chains, we establish some sufficient conditions for the analyticity and univalency of functions defined by an integral operator. Also, we refine the result to a quasiconformal extension criterion with the help of Beckers's method.
We describe the Loewner chains of the real locus of a class of real rational functions whose critical points are on the real line. Our main result is that the poles of the rational function lead to explicit formulas for the dynamical system that governs the driving functions. Our formulas give a simple method for mapping the class of rational functions into solutions to a non-trivial system of quadratic equations, and for directly showing that the curves in the real locus satisfy geometric commutation and have the geodesic multichord property. These results are entirely self-contained and have…
A well-known theorem by J. Becker states that if a normalized univalent function $f$ in the unit disk $\mathbb{D}$ can be embedded as the initial element into a Loewner chain $(f_t)_{t\geqslant 0}$ such that the Herglotz function $p$ in the Loewner -- Kufarev PDE $$\partial f_t(z)/\partial f=zf'_t(z)p(z,t),\qquad z\in\mathbb{D},\quad\mathrm{a.e.}~t\ge0,$$ satisfies $\big|(p(z,t)-1)/(p(z,t)+1)\big|\le k<1$, then $f$ admits a $k$-q.c. (="$k$-quasiconformal") extension $F:\mathbb{C}\to\mathbb{C}$. The converse is not true. However, a simple argument shows that if $f$ has a $q$-q.c. extension with…
