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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24 views

A quasiregular map from the half plane to the disc

Let $\mathbb{H}:=\{z\in\mathbb{C}:\operatorname{Im} z> 0\}$ be the upper half plane in $\mathbb{C}$. Let $N$ be the set of points in the real axis with real coordinates $\pm \log n, n\in\mathbb{N}$....
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25 views

Question about Hubbard's analytic definition of quasiconformality. Aren't weak derivatives only defined up to a set of measure zero?

I'm a bit confused about something that appears in the fourth chapter of Hubbard's Teichmüller Theory text. In his statement of Weyl's lemma, he says that if $f$ is a distribution whose weak/...
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40 views

Quasiconformal maps take null sets to null sets

Let $f:U\rightarrow V$ be some quasiconformal map in the plane. I want to show that the formula $$\mathrm{area}(f(E)) = \int_{E}|f_z|^2-|f_{\bar{z}}|^2\mathrm{d}x\, \mathrm{d}y$$ is valid. There are ...
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16 views

$p$-admissible weight, which is not an $A_p$ weight

Let $\Omega\subset\mathbb{R}^N$ be a bounded and smooth domain with $N\geq 2$. Let us consider the following class of weights $$ B_s=\{w\in W_p: w^{-s}\in L^1(\Omega)\text{ for some }s\in Y\},\quad 1&...
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22 views

Inverse conformal flattening

I am trying to implement a code for inverting the effect of conformal flattening and, as a consequence, interpolating from a flattened mesh in $\mathbb{R}^2$ to a surface lying in $\mathbb{R}^3$. The ...
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21 views

Find a bilipschitz map from the boundary of a bounded convex domain of $\mathbb{R}^2$ to $S^1$

Let $G\subseteq \mathbb{R^2}$ be a bounded convex domain. I need to find a bilipschitz map from $\partial G$ to $S^1$, i.e. a map $f:\partial G\rightarrow S^1$ that satisfies for all $x,y \in \partial ...
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67 views

Can we wrap a square onto itself with constant singular values?

I have now cross-posted this on mathoverflow. Let $0<\sigma_1<\sigma_2$ satisfy $\sigma_1\sigma_2=1$, and let $D=[-1,1]^2$. Does there exist a Lipschitz bijective* map $f:D \to D$ such that $df$...
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71 views

'Classical' Infinitesimals and Tangent Spaces

I do not know much differential geometry, and was led to this question from complex dynamics. It seems that it is often possible to reason 'infinitesimally' about maps between tangent spaces. For ...
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106 views

Conformal mapping of the angle $− \frac{π}{4} <Arg (z) <\frac{π}{2}$ to the right half plane $Im(w)> 0$

Could you help me with the following please: Find the conformal mapping of the angle $− \frac{π}{4} <Arg (z) <\frac{π}{2}$ to the right half plane $Im(w)> 0$ such that $w (1 - i) = 2, w (i) = ...
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94 views

Is a holomorphic function with nonvanishing derivative almost injective?

Let $\Omega \subseteq \mathbb C$, be an open, bounded, connected, contractible subset with smooth boundary. Let $f:\Omega \to \mathbb C$ be holomorphic, and suppose that its derivative $f'$ is ...
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156 views

Find a conformal map onto the unit disk

I am trying to find a conformal map from $G =\{re^{i\theta}| 0<r<1, \frac{-\pi}{2} < \theta < \pi\}$ onto the unit disk. I have an attempt but I am not sure if it is correct. Let $f_1(z) ...
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152 views

Can quasiconformal mappings converge uniformly to a homeomorphism that is NOT quasiconformal?

My question concerns the following situation: Let $G$ be a domain in $\mathbb{C}$ and $f_n: G \rightarrow \mathbb{C}$ be a sequence of quasiconformal mappings. Suppose that $f_n$ converges uniformly ...
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43 views

Conformal transformation

This is a problem in Volkovyskii's book. I have a little difficulty finding the right result. Who can show me how to find the conformal mapping how affects the exterior of the parabola $y^2=2px,\,\,(...
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21 views

Derivatives at the zero point of a conformal map from the unit strip onto the unit disc

Let $w$ be a point in the unit strip $\mathbb{S}=\{z\in \mathbb{C} : 0< Re\, z<1\}$ and let $\phi_w$ be a conformal map from $\mathbb{S}$ onto the unit disc $\mathbb{D}$ with $\phi_w(w)=0$. It ...
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Does there exist a volume-preserving diffeomorphism of the disk without conformal points?

This question is related to this one, though is supposed to be easier. Let $D \subseteq \mathbb{R}^2$ be the closed unit disk. Does there exist a smooth volume-preserving diffeomorphism $f:D \to ...
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Prove $\iint_\mathbb{D} \bar{\partial}f(z)(\zeta-z)^{-2}\mathrm dx\mathrm dy=\pi (f'(\zeta)-1)$

When I read the paper Teichmuller spaces and BMOA by K. Astala and M. Zinsmeister, I am stuck in the following eqution for a long time. $$\iint_\mathbb{D} \bar{\partial}f(z)(\zeta-z)^{-2}\mathrm dx\...
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87 views

Homeomorphism Between Closed Riemann Surfaces Homotopic to Quasiconformal Mapping

I'm re-reading a paper of Bers and for the second time, and I am yet again confused about the claim in the title, which Bers declares to be easy to prove. For context, I'll lay out some terminology....
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18 views

Finding a conformal mapping from one space to another

I am having trouble understanding how to go about finding a conformal mapping from one arbitrary space to another. I have made a few assumptions, but I am not sure if these are all correct. I assume ...
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Conformal representation [closed]

[Complex analysis question] I'm not sure if it is even possible to do conformal representation of function G to G*, since I need to transform one circle (line is circle with infinite radius) to two ...
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162 views

Quasiconformal Mappings - Definition of Modulus of a `Quadrilateral'

For context, I am studying background material for as well as the basics of Teichmuller theory. I am currently struggling to understand Lehto's definition of quasiconformal in his text "Quasiconformal ...
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167 views

Question on conformal mappings

The question: Let $\ Ω$ be a simply connected domain Let $\phi_{1}$ and $\phi_{2}$ be conformal self maps on $\ Ω$. Let $P, Q$ be distinct points in $\ Ω$ If $\phi_{1} (P) = \phi_{2}(P)$ and $\phi_{1}...
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Find the form of $f:\mathbb{C}\to\mathbb{C}$ which is entire, conformal and $\lim_{z\to\infty} f(z)=\infty$

Find the most general form of a function $f:\mathbb{C}\to\mathbb{C}$ which is entire, conforal and $\lim_{z\to\infty} f(z)=\infty$. I know that if $f$ is entire and $\lim_{z\to\infty}f(z)=\infty$ ...
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Exercice Quasiconformal Surgery (4.2.3)

I was tring to do this exercise from Branner and Fagella book on Quasiconformal Surgery. Suppose $P$ is a polynomial with a superattracting fixed point, say $\alpha$, whose immediate basin, $\mathcal{...
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228 views

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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77 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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166 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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59 views

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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81 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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156 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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944 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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170 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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158 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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158 views

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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54 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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108 views

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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121 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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213 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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114 views

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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255 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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60 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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114 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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56 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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364 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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36 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
169 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
150 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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129 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
97 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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292 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 ...