Loewner Chain Research Articles

Pole dynamics and an integral of motion for multiple SLE(0)

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. These results are entirely self-contained and have no reliance on probabilistic objects, but make use of an integral of motion for the Loewner chain that is motivated by ideas from conformal field theory. We also show that the dynamics of the driving functions are a special case of the Calogero–Moser integrable system, restricted to a particular submanifold of phase space carved out by the Lax matrix. Our approach complements a recent result of Peltola and Wang, who showed that the real locus is the deterministic κ→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa \\rightarrow 0$$\\end{document} limit of the multiple SLE(κ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\kappa )$$\\end{document} curves.

G−Loewner chain associated with Muir-type extension operator

In this paper, we will give two sufficient conditions that the Muir-type extension operator preserves the first element of a Loewner chain on the Reinhardt domain , where or . Using these conclusions, it can be easily proved that the Muir-type extension operator preserves starlikeness of order α and strong starlikeness.

Shearing maps and a Runge map of the unit ball which does not embed into a Loewner chain with range Cn

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 $\C^n$ which however cannot be embedded into a Loewner chain whose range is $\C^n$.

The level sets of functions with bounded criticalsets and bounded Hess+ complements

"We denote by ${\rm Hess}^+(f)$ the set of all points $p\in\mathbb{R}^n$ such that the Hessian matrix $H_p(f)$ of the $C^2$-smooth function $f:\mathbb{R}^n\longrightarrow\mathbb{R}$ is positive definite. In this paper we prove several properties of real-valued functions of several variables by showing the connectedness of their level sets for sufficiently high levels, under the boundedness assumption on the critical set. In the case of three variables we also prove the convexity of the levels surfaces for sufficiently high levels, under the additional boundedness assumption on the ${\rm Hess}^+$ complement. The selection of the {\em a priori} convex levels, among the connected regular ones, is done through the positivity of the Gauss curvature function which ensure an ovaloidal shape of the levels to be selected. The ovaloidal shape of a level set makes a diffeomorphism out of the associated Gauss map. This outcome Gauss map diffeomorphism is then extended to a smooth homeomorphism which is used afterwards to construct one-parameter families of smooth homeomorphisms of Loewner chain flavor. "

A criterion of univalence in Cn in terms of the Schwarzian derivative

"Using the Loewner Chain Theory, we obtain a new criterion of univalence in Cn in terms of the Schwarzian derivative for locally biholomorphic mappings. We as well derive explicitly the formula of this Schwarzian derivative using the numerical method of approximation of zeros due by Halley."

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.

Multiple backward Schramm–Loewner evolution and coupling with Gaussian free field

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 function, which is determined by a system of Belavin–Polyakov–Zamolodchikov-like equations so that these Loewner chains are commutative. After this observation, we define a multiple backward SLE as a tuple of mutually commutative backward Loewner chains. It immediately follows that each backward Loewner chain in a multiple backward SLE is obtained as a Girsanov transform of a backward SLE. We also discuss coupling of a multiple backward SLE with a GFF with boundary perturbation and find that a partition function and a boundary perturbation are uniquely determined so that they are coupled with each other.

Topological Loewner theory on Riemann surfaces

We prove that topological evolution families on a Riemann surface S are rather trivial unless S is conformally equivalent to the unit disc or the punctuated unit disc. We also prove that, except for the torus where there is no non-trivial continuous Loewner chain, there is a topological evolution family associated to any topological Loewner chain and, conversely, any topological evolution family comes from a topological Loewner chain on the same Riemann surface.

Loewner chains and nonlinear resolvents of the Carathéodory family on the unit ball in [formula omitted

In this paper we study various properties of nonlinear resolvents of holomorphic mappings in the Carathéodory family M(Bn), where Bn is the Euclidean unit ball in Cn. First, we prove certain characterizations of inverse Loewner chains f(z,t)=e−∫0ta(τ)dτz+⋯ on Bn×[0,∞), where a:[0,∞)→C is a locally Lebesgue integrable function such that ℜa(t)>0 for a.e. t≥0, and which satisfies a natural assumption. Next, we prove that if f∈M(Bn), then the nonlinear resolvent family {Jr}r≥0 is an inverse Loewner chain on Bn and the associated Herglotz vector field is of divergence type, where Jr=Jr[f]=(In+rf)−1, r≥0. This result is a generalization to higher dimensions of a recent result due to Elin, Shoikhet and Sugawa. We prove that (1+r)Jr can be embedded as the first element of a normal Loewner chain and the family {(1+r)Jr[f]:0≤r<∞,f∈M(Bn)} is a compact subset of S0(Bn). Also, we deduce that the shearing of (1+r)Jr (r≥0) associated with the family M(B2) is quasi-convex of type A and also starlike of order 4/5. We give a sufficient condition for (1+r)Jr to be quasiconformal on Bn and to be extended to a quasiconformal homeomorphism of Cn onto itself. Finally, sharp coefficient bounds for the nonlinear resolvent families of certain subsets of M(Bn), and also examples of support points of the compact families generated by nonlinear resolvent mappings on B2, will be obtained.

Operator-valued chordal Loewner chains and non-commutative probability

We adapt the theory of chordal Loewner chains to the operator-valued matricial upper-half plane over a C⁎-algebra A. We define an A-valued chordal Loewner chain as a subordination chain of analytic self-maps of the A-valued upper half-plane, such that each Ft is the reciprocal Cauchy transform of an A-valued law μt, such that the mean and variance of μt are continuous functions of t.We relate A-valued Loewner chains to processes with A-valued free or monotone independent increments just as was done in the scalar case by Bauer [8] and Schleißinger [48]. We show that the Loewner equation ∂tFt(z)=DFt(z)[Vt(z)], when interpreted in a certain distributional sense, defines a bijection between Lipschitz mean-zero Loewner chains Ft and vector fields Vt(z) of the form Vt(z)=−Gνt(z) where νt is a generalized A-valued law.Based on the Loewner equation, we derive a combinatorial expression for the moments of μt in terms of νt. We also construct non-commutative random variables on an operator-valued monotone Fock space which realize the laws μt. Finally, we prove a version of the monotone central limit theorem which describes the behavior of Ft as t→+∞ when νt has uniformly bounded support.

An embedding of the unit ball that does not embed into a Loewner chain

We construct a holomorphic embedding \(\phi :\mathbb B^3\rightarrow {\mathbb {C}}^3\) such that \(\phi ({\mathbb {B}}^3)\) is not Runge in any strictly larger domain. As a consequence, \(\mathcal S\ne {\mathcal {S}}^1\) for \(n=3\).

Loewner chain and quasiconformal extension of some classes of univalent functions

ABSTRACTIn 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 found by applying a well-known result.

The energy of a deterministic Loewner chain: Reversibility and interpretation via SLE$_{0+}$

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 _{\kappa} as \kappa \to 0 and the known reversibility of the SLE _{\kappa} curves for small \kappa , 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 .

Continuity of the Phase Transition for Planar Random-Cluster and Potts Models with $${1 \le q \le 4}$$ 1 ≤ q ≤ 4

This article studies the planar Potts model and its random-cluster representation. We show that the phase transition of the nearest-neighbor ferromagnetic $q$-state Potts model on $\mathbb Z^2$ is continuous for $q\in\{2,3,4\}$, in the sense that there exists a unique Gibbs state, or equivalently that there is no ordering for the critical Gibbs states with monochromatic boundary conditions. The proof uses the random-cluster model with cluster-weight $q\ge1$ (note that $q$ is not necessarily an integer) and is based on two ingredients: 1. The fact that the two-point function for the free state decays sub-exponentially fast for cluster-weights $1\le q\le 4$, which is derived studying parafermionic observables on a discrete Riemann surface. 2. A new result proving the equivalence of several properties of critical random-cluster models: - the absence of infinite-cluster for wired boundary conditions, - the uniqueness of infinite-volume measures, - the sub-exponential decay of the two-point function for free boundary conditions, - a Russo-Seymour-Welsh type result on crossing probabilities in rectangles with arbitrary boundary conditions. The result leads to a number of consequences concerning the scaling limit of the random-cluster model with $1\le q \le 4$. It shows that the family of interfaces (for instance for Dobrushin boundary conditions) are tight when taking the scaling limit and that any sub-sequential limit can be parametrized by a Loewner chain. We also study the effect of boundary conditions on these sub-sequential limits. Let us mention that the result should be instrumental in the study of critical exponents as well.

Extremal Problems for Mappings with Generalized Parametric Representation in $${{\mathbb {C}}}^{n}$$ C n

In this paper we are concerned with the family \(\widetilde{S}^t_A(\mathbb {B}^n)\) (\(t\ge 0\)) of normalized biholomorphic mappings on the Euclidean unit ball \(\mathbb {B}^n\) in \({\mathbb {C}}^n\) that can be embedded in normal Loewner chains whose normalizations are given by time-dependent operators \(A\in \widetilde{\mathcal {A}}\), where \(\widetilde{\mathcal {A}}\) is a family of measurable mappings from \([0,\infty )\) into \(L({\mathbb {C}}^n)\) which satisfy certain natural assumptions. In particular, we consider extreme points and support points associated with the compact family \(\widetilde{S}^t_A(\mathbb {B}^n)\), where \(A\in \widetilde{\mathcal {A}}\). We prove that if \(f(z,t)=V(t)^{-1}z+\cdots \) is a normal Loewner chain such that \(V(s)f(\cdot ,s)\in \mathrm{ex}\,\widetilde{S}^s_A(\mathbb {B}^n)\) (resp. \(V(s)f(\cdot ,s)\in \mathrm{supp}\,\widetilde{S}^s_A(\mathbb {B}^n)\)), then \(V(t)f(\cdot ,t)\in \mathrm{ex}\, \widetilde{S}^t_A(\mathbb {B}^n)\), for all \(t\ge s\) (resp. \(V(t)f(\cdot ,t)\in \mathrm{supp}\,\widetilde{S}^t_A(\mathbb {B}^n)\), for all \(t\ge s\)), where V(t) is the unique solution on \([0,\infty )\) of the initial value problem: \(\frac{d V}{d t}(t)=-A(t)V(t)\), a.e. \(t\ge 0\), \(V(0)=I_n\). Also, we obtain an example of a bounded support point for the family \(\widetilde{S}_A^t(\mathbb {B}^2)\), where \(A\in \widetilde{\mathcal {A}}\) is a certain time-dependent operator. We also consider the notion of a reachable family with respect to time-dependent linear operators \(A\in \widetilde{\mathcal {A}}\), and obtain characterizations of extreme/support points associated with these families of bounded biholomorphic mappings on \(\mathbb {B}^n\). Useful examples and applications yield that the study of the family \(\widetilde{S}^t_A(\mathbb {B}^n)\) for time-dependent operators \(A\in \widetilde{\mathcal {A}}\) is basically different from that in the case of constant time-dependent linear operators.

Loewner Chain Associated with the Modified Roper–Suffridge Extension Operator

In this paper, we will determine under what condition the modified Roper–Suffridge extension operators F and \(\Psi _{n,\frac{1}{p_2},\ldots ,\frac{1}{p_n}}(f)\) can be embedded in a Loewner chain on the unit ball \(B^n\) and on a special bounded convex circular domain \(\Omega _{n,p_2,\ldots ,p_n}\), respectively. In particular, some well-known results can be obtained using the main theorems in this paper.

Embedding univalent functions in filtering Loewner chains in higher dimension

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 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.

Shearing Process and an Example of a Bounded Support Function in $$S^0(\mathbb B^2)$$ S 0 ( B 2 )

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.

Some applications of variation of Loewner chains in several complex variables

In a recent paper, Bracci, Graham, Hamada, Kohr developed a new method to construct Loewner chains, by considering variations of certain Loewner chains. We use this method first, to prove a topological property of the class S0 of mappings with parametric representation on the Euclidean unit ball Bn, which for n≥2 immediately implies the density of the automorphisms of Cn that, restricted to Bn, have parametric representation, and second, to prove that every normalized univalent mapping on Bn whose image is Runge and which is C1 up to the boundary embeds into a normalized Loewner chain with range Cn.

Extreme points, support points and the Loewner variation in several complex variables

In this paper we consider extreme points and support points for compact subclasses of normalized biholomorphic mappings of the Euclidean unit ball Bn in ℂn. We consider the class S0(Bn) of biholomorphic mappings on Bn which have parametric representation, i.e., they are the initial elements f(·, 0) of a Loewner chain f(z, t) = etz + … such that {e−tf(·, t)}t⩾0 is a normal family on Bn. We show that if f(·, 0) is an extreme point (respectively a support point) of S0(Bn), then e−tf(·, t) is an extreme point of S0(Bn) for t ⩽ 0 (respectively a support point of S0(Bn) for t ∈ [0,t0] and some t0 > 0). This is a generalization to the n-dimensional case of work due to Pell. Also, we prove analogous results for mappings which belong to S0(Bn) and which are bounded in the norm by a fixed constant. We relate the study of this class to reachable sets in control theory generalizing work of Roth. Finally we consider extreme points and support points for biholomorphic mappings of Bn generated by using extension operators that preserve Loewner chains.