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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.

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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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1answer
63 views

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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26 views

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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22 views

conformal injective map

I have a conformal, injective map $f: G\subset \mathbb{C} \rightarrow \mathbb{C} $, G a domain and $D=\{ z \in \mathbb{C}: |z|\leq 1\} \subset G $. Furthermore $z_0$ is boundary point of D with $ |f(...
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Finding the area of this ellipse written in complex polar form.

I was reading the Chapter $4$ of Hubbard's Teichmüller Theory where he begins to introduce quasiconformal mappings. Here he writes a linear tranformation $T: \mathbb{C} \rightarrow \mathbb{C}$ as $T(u)...
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63 views

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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83 views

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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1answer
215 views

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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88 views

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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1answer
67 views

A counter example for Liouville's theorem when the Jacobian is changing signs

The famous Liouville's theorem states the following: Let $\Omega$ be a domain in $\mathbb{R}^n$, and let $f \in W_{loc}^{1,n}(\Omega,\mathbb{R}^n)$ satisfy $Jf=\det df \ge 0$ a.e. on $\Omega$ or $\...
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Question concerning Schwarz-Christoffel Mappings and Conformal Modulus

By the Riemann Mapping Theorem we know every region (open, connected subset of $\mathbb{C}$), that isn't the whole plane is conformally equivalent to the unit disk $\mathbb{D}$. By the Schwarz-...
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34 views

Construct a Mobius transformation $f$ with the specified effect:

Construct a Mobius transformation $f$ with the specified effect: $f$ maps $K(0,1)$ to itself and $K(1/4,1/4)$ to $K(0,r)$ for some $r<1$. My work: Since $i \rightarrow 1 \leftarrow i$ $ 1/4 \...
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Quasiregularity almost everywhere (removability)

The three equivalent definitions of quasiregular mapping that I am using are these ones: Let $U\subset\mathbb{C}$ be an open set and $K < \infty$. Then: A mapping $g:U\to\mathbb{C}$ is $...
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Convergence of complex dilatation of composed quasiconformal mappings

My question refers to the notion of good approximation of quasiconformal mappings: Let $G, G' \subseteq \mathbb{C}$ be domains. A sequence $(f_n)_n$ of quasiconformal mappings of $G$ onto $G'$ is ...
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1answer
54 views

Holomorphic mappings send sets of measure zero to sets of measure zero.

I want to see why holomorphic mappings send sets of measure zero to sets of measure zero. I found this statement reading about quasiconformal mappings. In fact, there is a theorem (see for instance L....
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101 views

Uniformly convergent sequence of quasiconformal mappings

Let $G \subseteq \mathbb{C}$ be a domain (one may assume that $G$ is bounded and simply connected, if needed). Suppose $(f_n)_{n \in \mathbb{N}}$ is a sequence of bounded (i.e. $\sup_{z \in G} |f_n(z)|...
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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 ...
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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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41 views

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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1answer
98 views

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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1answer
49 views

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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236 views

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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1answer
35 views

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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1answer
75 views

Line to hyperbola conformal transformation

Given the line $x^\alpha=(x,1/2)$ and the parameter $c^\alpha = (0,-1)$, I know that the transformation given by: $\tilde{x}_1 = \frac{x}{-x^2 + 1/4}$ $\tilde{x}_2 = \frac{x^2+\frac{1}{4}}{\...
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1answer
116 views

An inequality for conformal maps from Ahlfors

I'm reading "Lectures on Quasiconformal Mappings" by Lars Ahlfors. On page 16, while proving a lemma, he states two inequalities without justification. I would like to know why these inequalities hold....
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1answer
102 views

Quasiconformal automorphism group of domains

My problem refers to the theory of quasiconformal mappings in $\mathbb{C}$: Let $\emptyset \not = D \subseteq \mathbb{C}$ be a domain (i.e. open and connected subset) - for the sake of simplicity, ...
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1answer
71 views

Construction of a quasiconformal mapping of the closed disk to itself

My problem relates to the construction of the quasiconformal map $T:\overline{\mathbb{D}} \longrightarrow \overline{\mathbb{D}}$ from beginning of the proof of Lemma 2.2. in Marshall-Rhode. Question ...
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247 views

Proof explanation: Ahlfors' solution of Mori's extremal problem

The third extremal problem (due to Mori) presented in the Ahlfors' lectures on quasiconformal mappings is the following: Let $G$ be a doubly connected region in $\mathbb{C}$, and denote by $C_1$ the ...
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1answer
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$f$ conformal $\implies$ $f$ ACL?

In An Introduction to Teichmüller Spaces, by Imayoshi and Taniguchi, we have the following definition for an absolutely continuous function: where, I suppose, a function $g:I\to \mathbb{C}$, $I \...
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228 views

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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1answer
54 views

$r\rightarrow1/r$ 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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1answer
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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 ...
4
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1answer
81 views

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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2answers
106 views

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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1answer
132 views

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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374 views

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 ...
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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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1answer
116 views

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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1answer
264 views

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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1answer
228 views

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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1answer
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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 ...
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1answer
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Precise definition of conformal structure based on a Riemannian metric on a Riemann surface

As I read the literature, I keep having some doubt about what a " conformal structure on a Riemann surface " exactly means. ( You can assume all the Riemann surface in this literature have universal ...
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2answers
694 views

quasiconformal “automorphism” groups of julia sets

To motivate this question, let me begin with a picture: Each letter labels a "blob" of this quartic julia set. (is there a technical term for these parts?). Because of resolution limitations I haven'...