# Tagged Questions

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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### 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 ...
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### “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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### Analytical form of conformal mapping of simple closed curve

I want to create a mapping to map a simple closed curve to a unit disk, in analytic forms. The curve is simple enough, with several segments: each with an analytical form, but no general analytical ...
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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 PoincarĂ©-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 ...