Loewner Equation Research Articles

Geometric Description of Some Loewner Chains with Infinitely Many Slits

We study the chordal Loewner equation associated with certain driving functions that produce infinitely many slits. Specifically, for a choice of a sequence of positive numbers (bn)n≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(b_n)_{n\\ge 1}$$\\end{document} and points of the real line (kn)n≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(k_n)_{n\\ge 1}$$\\end{document}, we explicitily solve the Loewner PDE ∂f∂t(z,t)=-f′(z,t)∑n=1+∞2bnz-kn1-t\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\dfrac{\\partial f}{\\partial t}(z,t)=-f'(z,t)\\sum _{n=1}^{+\\infty }\\dfrac{2b_n}{z-k_n\\sqrt{1-t}} \\end{aligned}$$\\end{document}in H×[0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {H}\ imes [0,1)$$\\end{document}. Using techniques involving the harmonic measure, we analyze the geometric behaviour of its solutions, as t→1-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t\\rightarrow 1^-$$\\end{document}.

Dipolar Komatu–Loewner differential equation

By constructing a suitable holomorphic vector field for a finitely connected domain we derive that any Jordan arc in a strip standard slit domain D starting on the real axis can be described through dipolar Komatu–Loewner equation. The Komatu–Loewner equations for the slits of strip standard domain are established. We prove that conversely, the dipolar Komatu–Loewner equation generates a family of conformal maps from sub-domains of D onto another strip standard slit domains.

Drivers, hitting times, and weldings in Loewner's equation

AbstractIn addition to conformal weldings , simple curves growing in the upper half plane generate driving functions and hitting times through Loewner's differential equation. While the Loewner transform and its inverse have been carefully examined, less attention has been paid to the maps . We study the continuity properties of these latter transformations and show that uniform driver convergence implies uniform hitting time convergence and uniform welding convergence, even when the corresponding curves do not converge. Welding convergence implies neither hitting time nor driver convergence, while hitting time convergence implies driver convergence in (at least) the case of constant drivers. As an application, we show that a curve of finite Loewner energy can be well approximated by an energy minimizer that matches ’s welding on a sufficiently‐fine mesh.

Loewner time conversion for q-generalized stochastic dynamics

Generalized statistical mechanics based on q-Gaussian has been demonstrated to be an effective theoretical framework for the analysis of non-equilibrium systems. Since q-generalized (non-extensive) statistical mechanics reduces the nonlinearity in the system into deformed entropy and probability distributions, we introduce an alternative method based on the time conversion method using the Loewner equation by investigating the statistical physical properties of one-dimensional stochastic dynamics described by the Langevin equation with multiplicative noise. We demonstrate that a randomized time transformation using Loewner time enables the conversion of the multiplicative Langevin dynamics into an equilibrium system obeying a conventional microcanonical ensemble. For the equilibrium Langevin system after the Loewner time conversion, the fluctuation-dissipation relation and path integral fluctuation theorem were discussed to derive the response function under a nonlinear perturbation and an extended Jarzynski equality. The present results suggest the efficacy of the introducing randomized time for analyzing non-equilibrium systems, and indicate a novel connection between q-generalized (non-extensive) and Boltzmann–Gibbs statistical mechanics.

On the Search for Parameters of a Conformal Mapping from a Half-Plane to a Circular Polygon

The problem of constructing a conformal mapping from the upper half-plane to a circular polygon is considered. The preimages of the vertices of the polygon and accessory parameters are determined by the generalized method of P. P. Kufarev for determining the parameters in the Christoffel–Schwarz integral. The method is based on the Loewner equation with boundary normalization. The problem of constructing a mapping from a half-plane onto the exterior of a polygon with a boundary consisting of straight-line segments is solved separately. Examples of mappings whose parameters are found by the Kufarev method are given.

An Asymptotic Radius of Convergence for the Loewner Equation and Simulation of $$SLE_{\kappa }$$ Traces via Splitting

In this paper, we shall study the convergence of Taylor approximations for the backward Loewner differential equation (driven by Brownian motion) near the origin. More concretely, whenever the initial condition of the backward Loewner equation (which lies in the upper half plane) is small and has the form $Z_{0} = \varepsilon i$, we show these approximations exhibit an $O(\varepsilon)$ error provided the time horizon is $\varepsilon^{2+\delta}$ for $\delta > 0$. Statements of this theorem will be given using both rough path and $L^{2}(\mathbb{P})$ estimates. Furthermore, over the time horizon of $\varepsilon^{2-\delta}$, we shall see that "higher degree" terms within the Taylor expansion become larger than "lower degree" terms for small $\varepsilon$. In this sense, the time horizon on which approximations are accurate scales like $\varepsilon^{2}$. This scaling comes naturally from the Loewner equation when growing vector field derivatives are balanced against decaying iterated integrals of the Brownian motion. As well as being of theoretical interest, this scaling may be used as a guiding principle for developing adaptive step size strategies which perform efficiently near the origin. In addition, this result highlights the limitations of using stochastic Taylor methods (such as the Euler-Maruyama and Milstein methods) for approximating $SLE_{\kappa}$ traces. Due to the analytically tractable vector fields of the Loewner equation, we will show Ninomiya-Victoir (or Strang) splitting is particularly well suited for SLE simulation. As the singularity at the origin can lead to large numerical errors, we shall employ the adaptive step size proposed by Tom Kennedy to discretize $SLE_{\kappa}$ traces using this splitting. We believe that the Ninomiya-Victoir scheme is the first high order numerical method that has been successfully applied to $SLE_{\kappa}$ traces.

Fractional stochastic Loewner evolution and scaling curves

In two-dimensional statistical mechanics, the Stochastic Loewner equation, introduced by Oded Schramm, provides a powerful technique to study and categorize critical random curves and interfaces. For the first time, a new type of stochastic Loewner equation entitled fractional stochastic Loewner evolution (FSLE) has been proposed. We introduce a wide class of fractal curves using the fractional time series as the driving function of the Loewner equation and local fractional integrodifferential operators. We suggest that, in addition to fractal dimension estimations, the Hurst index of the driving function leads significant differences in the FSLE curves. This modification introduces a new approach to categorize different types of scaling curves based on the Hurst index of the driving function. Such formalizations appear to be appropriate for studying a wide range of two-dimensional curves seen in statistical mechanics and natural phenomena.

Driving function of the vertical slit

We consider the chordal Loewner equation and construct a family of vertical slits ?p (p > 0). We give the exact expression of its driving function ?, and its H?lder exponent near 0 in terms of p, which maps onto (1/2,?) and has a natural connection with the known results. In addition, we extend the asymptotic behavior of the driving function ? to a general case.

О решениях уравнения Лёвнера с составными управляющими функциями

О решениях уравнения Лёвнера с составными управляющими функциями

Field enhancement optimization of growing curved structures using the Loewner equation

The local electrostatic field enhancement factor (FEF) is analytically evaluated close to the tip of growing protrusions with quasi-one-dimensional curved geometries. This evaluation is done by using conformal transformations obtained as solutions to the Loewner equation under appropriate boundary conditions. This work allows us to detect the maximal FEF achieved during the growth of curved structures, providing theoretical support for its optimization, which is useful for the understanding of electrostatic interactions of complex free-form microstructured curved shapes.

Three phases of multiple SLE driven by non-colliding Dyson’s Brownian motions

The present paper is concerned with properties of multiple Schramm–Loewner evolutions (SLEs) labelled by a parameter κ ∈ (0, 8]. Specifically, we consider the solution of the multiple Loewner equation driven by a time change of Dyson’s Brownian motions in the non-colliding regime. Although it is often considered that several properties of the solution can be studied by means of commutation relations of SLEs and the absolute continuity, this method is available only in the case that the curves generated by commuting SLEs are separated. Beyond this restriction, it is not even obvious that the solution of the multiple Loewner equation generates multiple curves. To overcome this difficulty, we employ the coupling of Gaussian free fields and multiple SLEs. Consequently, we prove the longstanding conjecture that the solution indeed generates multiple continuous curves. Furthermore, these multiple curves are (i) simple disjoint curves when κ ∈ (0, 4], (ii) intersecting curves when κ ∈ (4, 8), and (iii) space-filling curves when κ = 8.

Remarks on the regularity of quasislits

A quasislit is the image of a vertical line segment \([0, iy]\), \(y > 0\), under a quasiconformal homeomorphism of the upper half-plane fixing \(\infty\). Quasislits correspond precisely to curves generated by the Loewner equation with a driving function in the Lip-\(\frac{1}{2}\) class. It is known that a quasislit is contained in a cone depending only on its Loewner driving function Lip-\(\frac{1}{2}\) seminorm, \(\sigma\). In this note we use the Loewner equation to give quantitative estimates on the opening angle of this cone in the maximal range \(\sigma <4\). The estimate is shown to be sharp for small \(\sigma\). As consequences, we derive explicit Hölder exponents for \(\sigma < 4\) as well as estimates on winding rates. We also relate quantitatively the Lip-\(\frac{1}{2}\) seminorm with the quasiconformal dilatation and discuss the optimal regularity of quasislits achievable through reparametrization.

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.

N-sided radial Schramm–Loewner evolution

We use the interpretation of the Schramm–Loewner evolution as a limit of path measures tilted by a loop term in order to motivate the definition of n-radial SLE going to a particular point. In order to justify the definition we prove that the measure obtained by an appropriately normalized loop term on n-tuples of paths has a limit. The limit measure can be described as n paths moving by the Loewner equation with a driving term of Dyson Brownian motion. While the limit process has been considered before, this paper shows why it naturally arises as a limit of configurational measures obtained from loop measures.

Tangential Loewner hulls

Through the Loewner equation, real-valued driving functions generate sets called Loewner hulls. We analyze driving functions that approach 0 at least as fast as \(a (T-t)^r\) as \(t \to T\), where \(r \in (0, 1/2)\), and show that the corresponding Loewner hulls have tangential behavior at time \(T\). We also prove a result about trace existence and apply it to show that the Loewner hulls driven by \(a(T-t)^r\) for \(r \in (0,1/2)\) have a tangential trace curve.

Exact Solutions of the Multiple Loewner Equation

We study integrability cases for the multiple Loewner differential equation which generates conformal mappings from the upper half-plane $$\mathbb{H}$$ of the complex plane with multiple slits onto $$\mathbb{H}$$ . The research is reduced to constant, square root and exponential driving functions of the Loewner equation. Moreover, conformal mappings from $$\mathbb{H}$$ minus symmetric circular curves emanating from the joint point at the origin, onto $$\mathbb{H}$$ , are represented as solutions to the multiple Loewner equation. The results supplement earlier descriptions for single slit mappings given by Kager, Nienhuis and Kadanoff.

Exact solutions of Loewner equations

We consider the exact singular solution of chordal Loewner equation and investigate a sequence of vertical slits $$\gamma ^p$$ , where $$p=4n+1$$ , $$n\in \mathbb {N}$$ . Our main result is to give an exact expression of the driving function $$\lambda $$ , and its Holder exponent near 0 in terms of p, which lies in (1/2, 1] and has a natural connection with the known results.

Geodesic Loewner paths with varying boundary conditions.

Equations of the Loewner class subject to non-constant boundary conditions along the real axis are formulated and solved giving the geodesic paths of slits growing in the upper half complex plane. The problem is motivated by Laplacian growth in which the slits represent thin fingers growing in a diffusion field. A single finger follows a curved path determined by the forcing function appearing in Loewner's equation. This function is found by solving an ordinary differential equation whose terms depend on curvature properties of the streamlines of the diffusive field in the conformally mapped 'mathematical' plane. The effect of boundary conditions specifying either piecewise constant values of the field variable along the real axis, or a dipole placed on the real axis, reveal a range of behaviours for the growing slit. These include regions along the real axis from which no slit growth is possible, regions where paths grow to infinity, or regions where paths curve back toward the real axis terminating in finite time. Symmetric pairs of paths subject to the piecewise constant boundary condition along the real axis are also computed, demonstrating that paths which grow to infinity evolve asymptotically toward an angle of bifurcation of π/5.

Loewner evolution of hedgehogs and 2-conformal measures of circle maps

AbstractLet f be a germ of a holomorphic diffeomorphism with an irrationally indifferent fixed point at the origin in ${\mathbb C}$ (i.e. $f(0) = 0, f'(0) = e^{2\pi i \alpha }, \alpha \in {\mathbb R} - {\mathbb Q}$ ). Pérez-Marco [Fixed points and circle maps. Acta Math.179(2) (1997), 243–294] showed the existence of a unique continuous monotone one-parameter family of non-trivial invariant full continua containing the fixed point called Siegel compacta, and gave a correspondence between germs and families $(g_t)$ of circle maps obtained by conformally mapping the complement of these compacts to the complement of the unit disk. The family of circle maps $(g_t)$ is the orbit of a locally defined semigroup $(\Phi _t)$ on the space of analytic circle maps, which we show has a well-defined infinitesimal generator X. The explicit form of X is obtained by using the Loewner equation associated to the family of hulls $(K_t)$ . We show that the Loewner measures $(\mu _t)$ driving the equation are 2-conformal measures on the circle for the circle maps $(g_t)$ .

Limits of Radial Multiple SLE and a Burgers–Loewner Differential Equation

We consider multiple radial SLE as the number of curves tends to infinity. We give conditions that imply the tightness of the associated processes given by the Loewner equation. In the case of equal weights, the infinite-slit limit is described by a Loewner equation whose Herglotz vector field is given by a Burgers differential equation. Furthermore, we investigate a more general form of the Burgers equation. On the one hand, it appears in connection with semigroups of probability measures on the unit circle with respect to free convolution. On the other hand, the Burgers equation itself is also a Loewner differential equation for certain subordination chains.