Related papers: Matching univalent functions and conformal welding
We study conformal mappings from the unit disk (or a rectangle) to one-tooth gear-shaped planar domains from the point of view of the Schwarzian derivative, with emphasis on numerical considerations. Applications are given to evaluation of…
We solve the classical conformal welding problem for a composition of two random homeomorphisms generated by independent Gaussian multiplicative chaos measures with small parameter values. In other words, given two such measures on the…
Conformal mapping may be the best-known topic in complex analysis. Any simply connected nonempty domain $\Omega$ in the complex plane ${{\mathbb{C}}}$ (assuming $\Omega\ne {{\mathbb{C}}}$) can be mapped bijectively to the unit disk by an…
This paper is devoted to the study of conformal maps of the unit disk $\mathbb{D}$ in the plane onto a bounded Jordan domain $G$. The main aim is to show that such a map is asymptotically symmetric if and only if $G$ is bounded by a…
We present a new systematic method to construct the conformal mapping from outside the unit disc to outside of a simply connected domain using the generalized polarization tensors. We also present some numerical results to validate…
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…
We show how to express a conformal map of a general two connected domain in the plane such that neither boundary component is a point to a representative domain which has the virtue of having an explicit algebraic Bergman kernel function.…
Let G be a bounded Jordan domain in the complex plane with piecewise analytic boundary. We present theoretical estimates and numerical evidence for certain phenomena, regarding the application of the Bergman kernel method with algebraic and…
It is shown that there is a computable conformal map of the unit disk onto a domain $D$ that has a computable extension to the closure of the unit disk even though the boundary of $D$ is not effectively locally connected. The proof encodes…
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 Szeg\H{o} kernel expansion. These polynomials converge to the conformal…
Given a $C^1$ homeomorphism of the unit circle $\gamma$, let $f$ and $g$ be respectively the normalized conformal maps from the unit disc and its exterior so that $\gamma= g^{-1}\circ f$ on the unit circle. In this article, we show that by…
The conjugate function method is an algorithm for numerical computation of conformal mappings for simply and multiply connected domains on surfaces. In this paper the conjugate function method, earlier used for simply connected domains, is…
In the early 1980's an elementary algorithm for computing conformal maps was discovered by R. K\"uhnau and the first author. The algorithm is fast and accurate, but convergence was not known. Given points z_0,...,z_n in the plane, the…
Given a bounded n-connected domain in the plane bounded by non-intersecting Jordan curves, and given one point on each boundary curve, L. Bieberbach proved that there exists a proper holomorphic mapping of the domain onto the unit disc that…
We study conformal mappings from the unit disk to one-toothed gear-shaped planar domains from the point of view of the Schwarzian derivative. Gear-shaped (or "gearlike") domains fit into a more general category of domains we call "pregears"…
The conjugate function method is an algorithm for numerical computation of conformal mappings for simply and doubly connected domains. In this paper the conjugate function method is generalized for multiply connected domains. The key…
The matching problem for a given Jordan curve in the complex plane asks to find two nonconstant functions, one analytic in the bounded complementary component of the curve and the other analytic in the unbounded complementary component of…
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…
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…
Surface parameterizations have been widely used in computer graphics and geometry processing. In particular, as simply-connected open surfaces are conformally equivalent to the unit disk, it is desirable to compute the disk conformal…