Questions tagged [quasiconformal-maps]

Quasiconformal maps are generalizations of conformal maps. They started out being used in Nevanlinna's value distribution theory but now form a fundamental component of geometric function theory. This tag is for questions related to QC maps.

45 questions
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Could someone explain to me what a Teichmuller Space is?

In the simplest terms possible, for someone who understands the basics of Manifolds, Topology, but barely any of the more complicated topics. I've been using the following: http://homeowmorphism.com/...
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working out what a conformal map does!

I am trying to work out what the following conformal maps do. The motivation for this is to know that what must the characteristics of lambda be so that resulting image is in the right half plane (we ...
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Complex Directional Derivative

I'm working on an expository talk using the text Geometric Group Theory by Drutu & Kapovich. On page 722 they give the formula for the directional derivative of $f$ in the direction $e^{i\alpha}$ ...
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Finding a suitable transformation function for the picture

I haven't taken complex analysis yet so there may be many words to sift through and not much concrete mathematical notation. I'll try my best though. Given a local Euclidean unit square grid how ...
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Producing isothermal coordinates from a solution of the Beltrami Equation

Suppose that $f \colon U \to V$ is a diffeomorphism of planar domains. Its differential $Df$ can be pointwise expressed the sum of a complex-linear and complex-anti-linear mapping: Given a tangent ...
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A special Mobius Transformation that maps the right half plane to the unit disc

Find the Mobius Transformation that maps the right half plane to the unit disc carrying the point $z=15$ to the origin. Since the Mobius transformation takes the point $z=15$ to the origin, so I ...
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A conformal map whose Jacobian vanishes at a point is constant?

Let $f:M \to N$ be a smooth weakly conformal map between connected $d$-dimensional Riemannian manifolds, i.e. $f$ satisfies $df^Tdf =(\det df)^{\frac{2}{d}} \, \text{Id}$. Assume $d \ge 3$ and ...
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conformal mapping and rational function

Let $E$ be an infinite compact subset of the complex plane $\mathbb{C}$ such that $\overline{\mathbb{C}}\setminus E$ is simply connected. By Riemann mapping theorem, there exists a unique exterior ...
117 views

Book on quasiconformal mappings?

I am looking an introductory book on "quasiconformal mappings" for self-study. Also I would like to know about motivation and history behind this concept (I am a beginner of this subject). I really ...
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How to show this mapping is quasiconformal. And the integrability of the gradient.

A complex map $f$ on the unit disk defined as $f(re^{i\theta})=r^ke^{i\theta}$ where $k>1$. I hope to know how to show this map is quasiconformal and what is the largest $p$ such that the gradient ...
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Cauchy sequence of quasi-conformal automorphisms --> Conclusion on corresponding sequence of maximal dilatation?

Let $G \subsetneq \mathbb{C}$ be a simply connected, bounded domain in $\mathbb{C}$. Denote by $Q(G)$ the set of all quasi-conformal automorphisms of $G$, i.e. Q(G) := \left\{ f: G \rightarrow G \, ...
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Lines that don't pass through origin, map$(f(z)=\frac{1}{z})$ to disks. How do I map the line$(1,t); t\in (-\infty, \infty)$ to a disk?

Lines that don't pass through origin, map$(f(z)=\frac{1}{z})$ to disks. How do I map the line$(1,t); t\in (-\infty, \infty)$ to a disk? How about the general case $(t,at+b)$, $t\in(-\infty, \infty)$?...
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whats the image of this circle under the reciprocal mapping?

I need to find the image of $D(a,|a|)$ $a \in \mathbb{R}-\{0\}$ under the transformation $f(z) = 1/z$. I was trying write $z = |a|e^{i\theta} + a$ and then $e^{i\theta}=\cos(\theta) + i \sin(\theta)$ ...
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How can i find this transformation

i need to find a transformation $w= az+b$ the takes the triangle $1,i,0$ to the triangle $0,2,1+i$. I tried to use the fact the this transformation must take vertices to vertices but i get to a ...
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What is Extremal Length?

The question is as the title asks. My background is in computer science, and recently I'm trying to read a paper that involves using extremal length to prove certain properties of planar graphs. I ...
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$r\rightarrow1/r$ invariant

(Not sure the tags are appropriate, but can't think of better ones. Please suggest better.) Suppose you have a function $f(x,y,z,...;g(r))$ with the requirement that $r\rightarrow1/r$ leaves $f$ ...
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Looking for a biholomorphism [closed]

Is there a biholomorphism between $\mathbb{H}_2 \times S^1$ and a half-space of $\mathbb{R}^3$?
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Weakly quasisymmetric maps of a connected doubling space are quasisymmetric

I'm currently reading through a few chapters of Juha Heinonen's Lectures on Analysis on Metric Spaces, and I'm having some trouble understanding the finer points of a particular proof. The result is ...
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Locally quasiconformal implies quasiconformal

In Ahlfors' "Lectures on Quasiconformal Mappings," he shows that, using the geometric definition of K-quasiconformal, if a map between regions is locally K-quasiconformal then it is globally K-...
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Quasiconformal mappings: Metric deffinition

In the lectures notes http://users.jyu.fi/~pkoskela/quasifinal.pdf (Prof. Koskela has made them freely available from his webpage, so I am guessing is OK that I paste the link here) Quasiconformality ...
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Quasiconformal Mappings in Fluid Dynamics

I know that conformal mappings can be used to study 2 dimensional fluid flows. But I was wondering how quasiconformal mapping have been applied in this respect?
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Simplify Laplace equation in rectangle geometry

Consider Laplace's equation in a rectangle as shown in the following figure. The boundary conditions are shown in the figure. The problem is solved in the case of a1 =a2=1. Is there a way to ...
631 views

“Angle-preserving” equivalent to conformal?

I'd like to investigate the common turn of phrase that conflates "angle-preserving map" with "conformal map". Let $f:\Bbb R^2\to\Bbb R^2$ be a continuous function. I'll define $f$ to be angle-...
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Existence of certain analytic functions

Let $D_1,D_2\subset\mathbb{C}$ be two disjoint disks. Is there an analytic function defined on the upper half plane $f:\mathbb{H}\rightarrow\mathbb{C}$ such that $f$ takes each value in $D_1$ exactly ...
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comformal surface parameterization of simply-connected Riemann surface with boundary

$M$ and $S$ are two simply-connected surfaces with boundaries $\partial M =\partial S \neq \emptyset$, respectively. Both $M$ and $S$ are disk-like and smooth enough. Here, we assume $M$ is planar (...
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Quasiconformal map between the complex plane and a disk

According to the Poincaré-Koebe theorem, it is known that the unit disk $\mathbb D$ and the complex plane $\mathbb C$ aren't conformally equivalent. My question is maybe naive, but I was wondering if ...
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Lift of a homeomorphism $f$ between two (hyperbolic) surfaces $X,Y$

Let $X,Y$ be two hyperbolic Riemann surfaces (i.e. they have universal cover the upper half plane $\mathbb{H}$). Let $\pi_X:\mathbb{H}\to X, \pi_Y:\mathbb{H}\to Y$ be the corresponding covering maps. ...
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Give a example about invariant ergodic measure and quasi-symmetric mapping

Is there a example $(X,f,\mu)$ such that $X$ is a closed subset of Euclidean space, $f$ be a quasi-symmetric mapping but not a Lipschitz mapping, $f(X)=X$, $\mu$ is a finite measure on $X$ that is ...