Christopher J. Bishop
Given any $ε>0$ and any planar region $Ω$ bounded by a simple n-gon $P$ we construct a ($1 + ε)$-quasiconformal map between $Ω$ and the unit disk in time $C(ε)n$. One can take $ C(ε) = C + C \log (1/ε) \log \log (1/ε)$.
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We consider Riemann mappings from bounded Lipschitz domains in the plane to a triangle. We show that in this case the Riemann mapping has a linear variational principle: it is the minimizer of the Dirichlet energy over an appropriate affine space. By discretizing the variational principle in a natural way we obtain discrete conformal maps which can be computed by solving a sparse linear system. We show that these discrete conformal maps converge to the Riemann mapping in $H^1$, even for non-Delaunay triangulations. Additionally, for Delaunay triangulations the discrete conformal maps converge …
We propose an analytical approach to the conformal mapping of (rectangular) polygons based on the theory of Riemann surfaces and theta functions.
The conformal mapping $f(z)=(z+1)^2 $ from $\mathbb{D}$ onto the standard cardioid has a homeomorphic extension of finite distortion to entire $\mathbb{R}^2 .$ We study the optimal regularity of such extensions, in terms of the integrability degree of the distortion and of the derivatives, and these for the inverse. We generalize all outcomes to the case of conformal mappings from $\mathbb{D}$ onto cardioid-type domains.
Hayman's conjecture about covering of vertical intervals under regular mappings is proved.
In the article we consider the composite conformal map which maps annulus to infinite region with symmetric hole and nearly circular hole. It is shown that such transformation is good if the distance between centers of holes are large or radius of circular hole is small. Examples for bilinear-hypotrochoids mapping and bilinear-Schwarz-Christoffel mapping are present.
We prove necessary and sufficient integral conditions involving extremal distance for a conformal mapping of the unit disk to belong to the Hardy or weighted Bergman spaces. We also give characterizations for the Hardy number and the Bergman number of a simply connected domain in terms of extremal distance.
We present a new and simple proof of Teichmüller-Wittich-Belinskii's and Gutlyanskii-Martio's theorems on the conformality of quasiconformal mappings at a given point. Known proofs gave separate estimates for the radial and angular variations, but our proof unifies them using Grötzsch-type inequality for the variation of cross-ratio of four points on the Riemann sphere. We also give a sufficient condition for $C^{1+α}$-conformality
We extend the parameterization of sine-type functions in terms of conformal mappings onto slit domains given by Eremenko and Sodin to the more general case of generating functions of real complete interpolating sequences. It turns out that the cuts have to fulfill the discrete Muckenhoupt condition studied earlier by Lyubarskii and Seip.
This paper presents a MATLAB toolbox for computing the conformal mapping from a given polygonal multiply connected domain onto a circular multiply connected domain and its inverse. The toolbox can be used for multiply connected domains with high connectivity and complex geometry. It can be employed also for simply connected domains.
Analytical Error Estimation of Conformal Mappings Using Complex Bessel Functions under Perturbed Boundaries
本文研究了随机扰动边界下复Bessel函数保角映射的构造与误差估计，证明了映射的存在唯一性，推导了解析误差公式，并验证了其在小扰动条件下的渐近收敛性与稳定性。
Numerical Conformal Mapping to One-Tooth Gear-Shaped Domains and Applications
本文从Schwarzian导数视角研究单位圆盘或矩形到单齿齿轮形区域的数值共形映射，并应用于奇异积分求值、环形矩形补集映射及对称多齿区域。
M.Gromov extended the concepts of conformal and quasiconformal mapping to the mappings acting between the manifolds of different dimensions. For instance, any entire holomorphic function $ f: \Cn \to {\mathbb C}$ defines a mapping conformal in the sense of Gromov. In this connection Gromov addressed a natural question: which facts of the classical theory apply to these mappings? In particular is it true that {\em If the mapping $ F: \R^{n + 1} \to \R^{n}$ is conformal and bounded, then it is a constant mapping, provided that $ n \geq 2 $}~? We present arguments confirming the validity of such …
Conformal mapping of circular triangles & classical function theory
本文研究M.邦克已用解析与经典特殊函数方法详尽考察的特殊圆三角形的共形映射的几何定义与研究可能性，其研究方式与兼具解析和几何定义的椭圆模函数类似。
Conformal mappings on the Grushin plane
本文研究Grushin平面上的共形映射，通过Sobolev映射和几何性质给出刻画，并借助Meyerson映射与复平面共形性关联，应用于容许曲线、长度畸变估计及Carathéodory延拓定理。
We give an algorithm for finding conformal mappings onto the upper half-plane and conformal modules of some types of polygons. The polygons are obtained by stretching along the real axis polyominoes i.e., polygons which are connected unions of unit squares with vertices from the integer lattice. We consider the polyominoes of two types, so-called the $P$-pentomino and the $L$-tetromino. The proofs are based on the Riemann-Schwarz reflection principle and uniformization of compact simply-connected Riemann surfaces by rational functions.
Here we construct the conformal mappings with the help of continuous fractions approximations. These approximations converge to the algebraic roots $\sqrt[N]{z}$ for $N \in \mathbb{N}$ and $z$ from the right half-plane of the complex plane. We estimate both the convergence rate and the compact set of convergence. Also we give the examples that illustrate the introduced technique of a conformal mapping construction.
We define a new type of Hardy-Orlicz spaces of conformal mappings on the unit disk where in place of the value |f(x)| we consider the intrinsic path distance between f(x) and f(0) in the image domain. We show that if the Orlicz function is doubling then these two spaces are actually the same, and we give an example when the intrinsic Hardy-Orlicz space is strictly smaller.
We prove that a conformal mapping defined on the unit disk belongs to a weighted Bergman space if and only if certain integrals involving the harmonic measure converge. With the aid of this theorem, we give a geometric characterization of conformal mappings in Hardy or weighted Bergman spaces by studying Euclidean areas. Applying these results, we prove several consequences for such mappings that extend known results for Hardy spaces to weighted Bergman spaces. Moreover, we introduce a number which is the analogue of the Hardy number for weighted Bergman spaces. We derive various expressions f…
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.
Analytic Characterization of $t$-Quasicircles and Conformal Mappings onto $t$-Quasidisks
本文提出$(ρ,t)$-准对称映射，解析刻画$t$-准圆，并扩展了准圆定理至$t$-准圆的边界值性质。
Conformal Mappings Through the Lens of Invariant Metrics
本文证明了双曲平面区域上不变度量下的球是有限连通的，并以此为工具，为平面区域共形映射的经典结论提供了新的简明证明。
We study the uniform approximation of the canonical conformal mapping, for a Jordan domain onto the unit disk, by polynomials generated from the partial sums of the Szegő kernel expansion. These polynomials converge to the conformal mapping uniformly on the closure of any Smirnov domain. We prove estimates for the rate of such convergence on domains with piecewise analytic boundaries, expressed through the smallest exterior angle at the boundary. Furthermore, we show that the rate of approximation on compact subsets inside the domain is essentially the square of that on the closure. Two standa…
Conformal Mapping of Circular Quadrilaterals and Weierstrass Elliptic Functions
本文研究圆盘到轴对称圆弧四边形的共形映射，数值求解附属参数与几何参数的关联，结合含Weierstrass P-函数的矩形映射问题，得到P函数零点及半周期像的新公式。
Two triangle meshes are conformally equivalent if for any pair of incident triangles the absolute values of the corresponding cross-ratios of the four vertices agree. Such a pair can be considered as preimage and image of a discrete conformal map. In this article we study discrete conformal maps which are defined on parts of a triangular lattice $T$ with strictly acute angles. That is, $T$ is an infinite triangulation of the plane with congruent strictly acute triangles. A smooth conformal map $f$ can be approximated on a compact subset by such discrete conformal maps $f^\varepsilon$, defined …
A holomorphic function f on a simply connected domain Ω is said to possess a universal Taylor series about a point in Ω if the partial sums of that series approximate arbitrary polynomials on arbitrary compacta K outside Ω (provided only that K has connected complement). This paper shows that this property is not conformally invariant, and, in the case where Ω is the unit disc, that such functions have extreme angular boundary behaviour.
A new analytical method for the conformal mapping of a rectangular heptagon with a straight angle at infinity to a half plane and back is proposed. The method is based on the observation that SC integral in this case is an abelian integral on a genus two curve, so it may be represented in terms of Riemann theta functions. The approach is illustrated by the computation of 2D flow of ideal fluid above rectangular underlying surface and the computation of the capacities of multi component rectangular condensers with axial symmetry.
New algorithms are presented for numerical conformal mapping based on rational approximations and the solution of Dirichlet problems by least-squares fitting on the boundary. The methods are targeted at regions with corners, where the Dirichlet problem is solved by the "lightning Laplace solver" with poles exponentially clustered near each singularity. For polygons and circular polygons, further simplifications are possible.
The paper proposes a vector generalization of the basic concepts of the theory of complex variable: the concept of modulus and argument of complex number. The author introduces some generalizations of the notion of holomorphic functions and mappings to the case of infinite-dimensional complex spaces. This approach allows us to generalize several well-known results of geometric function theory to the case of infinite-dimensional complex spaces. In particular, the author summarizes a number of known theorems on the theory of functions of class S on infinite-dimensional complex space.
Harmonic Shears and Numerical Conformal Mappings
本文提出一种利用Clunie和Sheil-Small在1984年引入的剪切构造来寻找调和映射的数值算法，并讨论了与调和映射相关的最小曲面。
We study numerical conformal mappings of planar Jordan domains with boundaries consisting of finitely many circular arcs and compute the moduli of quadrilaterals for these domains. Experimental error estimates are provided and, when possible, comparison to exact values or other methods are given. The main ingredients of the computation are boundary integral equations combined with the fast multipole method.
We study the so-called integral means spectrum for univalent functions on the unit disk. Using an inequality of Prawitz (generalizing the classical area theorem of Gronwall), we find -- by applying a Moebius mapping to lift the result to the bidisk -- that it is possible to obtain rather strong estimates from above on the integral means spectrum.
In this note, we give an estimate for the dimension of the image of the unit circle under a quasiconformal mapping whose dilatation has small support. We also prove an analogous estimate for the rate of growth of a solution of a second-order parabolic equation given by the Feynman-Kac formula (with a sparsely supported potential) and introduce a dictionary between the two settings.
This paper presents a new uniquely solvable boundary integral equation for computing the conformal mapping, its derivative and its inverse from bounded multiply connected regions onto the five classical canonical slit regions. The integral equation is derived by reformulating the conformal mapping as an adjoint Riemann-Hilbert problem. From the adjoint Riemann-Hilbert problem, we derive a boundary integral equation with the adjoint generalized Neumann kernel for the derivative of the boundary correspondence function $θ'$. Only the right-hand side of the integral equation is different from a ca…
Conjugate Function Method for Numerical Conformal Mappings
提出基于共轭调和函数和四边形性质的方法，数值计算单连通或双连通区域到矩形或环形规范区域的共形映射。
Mappingsofbi-conformalenergyformthewidestclass of homeomorphisms that one can hope to build a viable extension of Geometric Function Theory with connections to mathematical models of Nonlinear Elasticity. Such mappings are exactly the ones with finite conformal energy and integrable inner distortion. It is in this way, that our studies extend the applications of quasiconformal homeomorphisms to the degenerate elliptic systems of PDEs. The present paper searches a bi-conformal variant of the Riemann Mapping Theorem, focusing on domains with exemplary singular boundaries that are not quasiballs.…
The conformal mapping problem for the section of a channel filled with porous material under a rectangular dam onto the upper half-plane is considered. Similar problems arise in computing of fluid flow in hydraulic structures. As a solution method, the representation of Christoffel-Schwartz elliptic integral in terms of Weierstrass functions is used. The calculation is based on Taylor series for the sigma function, the coefficients of which are determined recursively. A simple formula for a conformal mapping is obtained, which depends on four parameters and uses the sigma function. A numerical…
Quantitative stability of extremal quasi conformal mappings
本文研究了准共形映射理论中经典扭曲最小化问题的定量稳定性，证明了均值扭曲功能在特定边界条件下线性拉伸和螺旋拉伸映射的最优稳定性估计。
In this paper, we provide a candidate solution to obtain the coefficients of the conformal mapping for a lower half plane containing a symmetrical noncircular cavity using penalty function method and modified Particle Swarm Method. The nonconvexity of the penalty function is proven via the concept of convex function and proof of contradiction. The solution procedure is presented very detailedly in pseudocodes to ensure that the solution can be fully repeated and further improved. The solution accuracy and efficiency are also discussed.
Numerical conformal mapping
本文介绍数值共形映射的基本理论，阐述单连通、双连通及高连通区域到单位圆盘、圆环和典型区域的共形映射方法，并说明共形模的唯一确定性。
Laplace--Beltrami Equations and Numerical Conformal Mappings on Surfaces
本文扩展共轭函数法至黎曼曲面，利用hp自适应有限元求解Laplace--Beltrami方程，实现含奇异性和尖点复杂边界域的高精度共形映射计算。
Consider discrete conformal maps defined on the basis of two conformally equivalent triangle meshes, that is edge lengths are related by scale factors associated to the vertices. Given a smooth conformal map $f$, we show that it can be approximated by such discrete conformal maps $f^ε$. In particular, let $T$ be an infinite regular triangulation of the plane with congruent triangles and only acute angles (i.e.\ $<π/2$). We scale this tiling by $ε>0$ and approximate a compact subset of the domain of $f$ with a portion of it. For $ε$ small enough we prove that there exists a conformally equivale…
Bidirectional conformal mapping for over-break and under-break tunnelling and its application in complex variable method
针对不对称轮廓的超挖与欠挖隧道，提出结合电荷模拟法的双向共形映射，仅需一对线性系统，计算高效且易编码，并通过数值算例及与有限元解对比验证了其有效性。
On the Conformal Energy of Quasisymmetric and Quasimöbius Mappings
本文研究了有限扭曲极值映射的共形能量，直接根据数据给出了圆同胚共形能量的显式公式，证明了拟莫比乌斯自同胚的精确能量估计，并展示了有限共形能量圆同胚的一致逼近方法。
Beltrami coefficient and angular distortion of discrete geometric mappings
本文研究了Beltrami系数与离散几何映射角变形之间的关系，推导了理论结果，并通过数值实验验证了角变形的估计方法。
The Loewner-Kufarev evolution produces asymptotics for mappings onto domains close to the unit disk or the exterior of the unit disk. We deduce variational formulae which lead to the asymptotic conformal welding for such domains. The comparison of mappings onto bounded and unbounded components of the Jordan curve establishes an asymptotic connection between driving functions in both versions of the Loewner-Kufarev equation and conformal radii of the two domains.
The approximate conformal mapping of a disk onto domain with an acute angle
本文提出基于边界曲线重新参数化、Fredholm方程求解与样条插值的方法，构造单位圆盘到含尖角点有限区域的近似共形映射，并推广至多连通区域。
One Parameter Families of Conformal Mappings of Bounded Doubly Connected Polygonal Domains
本文基于参数化Loewner--Komatu方法，提出一种求圆环到有界双连通多边形区域共形映射的近似方法，通过推导描述附加参数与共形模量动力学的偏微分方程组，实现对多条割缝变化的完全控制，并用数值实验验证了方法的有效性。
Among various methods of finding accessory parameters in the Schwarz-Christoffel integrals, Kufarev's method, based on the Loewner differential equation, plays an important role. It is used for describing one-parameter families of functions that conformally map a canonical domain onto a polygon with a slit the endpoint of which moves along a polygonal line starting from a boundary point. We present a modification of Kufarev's method for the case of several slits, the lengths of which have depend of each other in a certain way. We justify the method and find a system of ODEs describing the dyna…
Given a conformal mapping $f$ of the unit disk $\mathbb D$ onto a simply connected domain $D$ in the complex plane bounded by a closed Jordan curve, we consider the problem of constructing a matching conformal mapping, i.e., the mapping of the exterior of the unit disk $\mathbb D^*$ onto the exterior domain $D^*$ regarding to $D$. The answer is expressed in terms of a linear differential equation with a driving term given as the kernel of an operator dependent on the original mapping $f$. Examples are provided. This study is related to the problem of conformal welding and to representation of …
In this paper we revisit the concept of conformality in the sense of Gauss in the context of octonions and Clifford algebras. We extend a characterization of conformality in terms of a system of partial differential equations and differential forms using special orthonormal sets of continuous functions that have been used before in the particular quaternionic setting. The aim is to describe to which higher dimensional algebras this characterization can exactly be extended and under which circumstances. It turns out to be crucial that this characterization requires a domain of definition that l…
On the Conformal Lens Map
本文证明了透镜映射具有特定阶的凸性和星形性，并给出了以开角表示的精确阶数。
This is a commentary on Teichm{ü}ller's paper Unter-suchungen{ü}ber konforme und quasikonforme Abbildungen (Inves-tigations on conformal and quasiconformal mappings) published in 1938. The paper contains fundamental results in conformal geometry , in particular a lemma, known as the Modulsatz, which insures the almost circularity of certain loci defined as complementary components of simply connected regions in the Riemann sphere, and another lemma, which we call the Main Lemma, which insures the circularity near infinity of the image of circles by a qua-siconformal map. The two results find w…
We examine several characteristics of conformal maps that resemble the variance of a Gaussian: asymptotic variance, the constant in Makarov's law of iterated logarithm and the second derivative of the integral means spectrum at the origin. While these quantities need not be equal in general, they agree for domains whose boundaries are regular fractals such as Julia sets or limit sets of quasi-Fuchsian groups. We give a new proof of these dynamical equalities. We also show that these characteristics have the same universal bounds and prove a central limit theorem for extremals. Our method is ba…
Convergence of the Zipper algorithm for conformal mapping
本文证明了Zipper算法（一种计算共形映射的初等方法）对于Jordan区域在一致边界逼近意义下的收敛性，并给出了映射函数及其逆在闭区域和闭圆盘上的一致估计。
We recall the notions of conformal and quasiconformal mappings \textit{in the sense of Gromov}, extending the classical notions of conformal and quasiconformal mappings, and prove the following theorem. {\em If the mapping $ F: \mathbb{R}^{n} \to \mathbb{R}^{2} $, where $ n \geq 2 $, quasiconformal in the sense of Gromov, omits more than one value on the plane $\mathbb{R}^{2} $, then it is a constant mapping.}
There are three types of results in this paper. The first, extending a representation theorem on a conformal mapping that omits two values of equal modulus. This was due to Brickman and Wilken. They constructed a representation as a convex combination with two terms. Our representation constructs convex combinations with unlimited number of terms. In the limit one can think of it as an integration over a probability space with the uniform distribution. The second result determines the sign of $\Re L(\overline{z}_0(f(z))^2)$ up to a remainder term which is expressed using a certain integral tha…
On the properness of $p$-conformal energy on the Teichmüller space of a Riemann surface
该研究证明了p-共形能量在固定亏格黎曼曲面的泰希米勒空间上为正当泛函，其核心依赖于对双曲环面间有限畸变唯一极值映射及其p-共形能量的显式识别结果。
Conformal mapping, a classical topic in complex analysis and differential geometry, has become a subject of great interest in the area of surface parameterization in recent decades with various applications in science and engineering. However, most of the existing conformal parameterization algorithms only focus on simply-connected surfaces and cannot be directly applied to surfaces with holes. In this work, we propose two novel algorithms for computing the conformal parameterization of multiply-connected surfaces. We first develop an efficient method for conformally parameterizing an open sur…
Let $D$ and $Ω$ be Jordan domains with Dini's smooth boundaries and and let $f:D\mapsto Ω$ be a harmonic homeomorphism. The object of the paper is to prove the following result: If $f$ is quasiconformal, then $f$ is Lipschitz. This extends some recent results, where stronger assumptions on the boundary are imposed, and somehow is optimal, since it coincides with the best condition for Lipschitz behavior of conformal mappings in the plane and conformal parametrization of minimal surfaces.
Efficient Numerical Conformal Mappings on Multiply Connected Riemann Surfaces
本文推广并改进了共轭函数法，用于多连通平面区域和黎曼曲面上的数值共形映射计算，通过高阶有限元方法实现了高精度，并推导了互逆误差估计。
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