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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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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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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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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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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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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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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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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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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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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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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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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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, ...
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 ...
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 ...