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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Quasiconformal maps
currently I am studying quasiconformal maps and the following question came up during my studies. In complex analysis one learns that
$$
\{f \colon \mathbb C \to \mathbb C : f \text{ is conformal }\} =...
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Conformal mapping of a doubly-connected polygonal domain onto an annulus
Currently, I have a domain D that is bounded by two polygons $C_1$ and $C_2$ (with $C_2$ completely 'inside' $C_1$). Now I want to map this domain onto $D'$ with the outer circle $C_1'$ (with unit ...
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What will happen with $T(x,y)$ if there is an injective conformal mapping $f$ that maps $(x, y)$ into $(u, v)$?
Currently, I have solved the temperature equation and I obtained the values $T(x, y)$ in the xy-space. I would like to map the xy-space onto the uv-space conformally and injective (that means that a ...
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Quasi-isometries forming a compact subset of mappings
I am reading the paper Un groupe hyperbolique est déterminé par son bord, and I am stuck on theorem 3.2, which says the following:
Theorem 3.2: Given two quasi-homogeneous hyperbolic spaces, $X$ and $...
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Measurable Riemann Mapping Theorem on a simply connected set
For context I am working through Sullivan's proof for the No-Wandering domain theorem.
My question is, can you restrict that Measurable Riemann Mapping Theorem to functions that are not defined on the ...
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Has the composition of a sequence of quasiconformal mappings with unbounded dilatation and another q.c. mapping still unbounded dilatation?
Let $G \subseteq \mathbb{C}$ be a bounded domain in $\mathbb{C}$. Consider for $m \in \mathbb{N}$ a sequence of quasiconformal mappings $f_m: G \longrightarrow \mathbb{C}$ with unbounded maximal ...
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Conformal map between simple and doubly connected domains
I know that, according to the Riemann mapping theorem, for any non-empty simply connected open subset $U$ of the complex plane $\mathbb{C}$ which is not all of $\mathbb{C}$, then there exists a ...
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Why joukowski transformation maps every circle passing through both -1 and +1 onto a curved segment?
want to show algebraically that ''Joukowski transformation''
$$w = z + {1 \over z}$$
maps every circle passing through both -1 and +1 onto a curved segment or arc. if a circle only passes through z=-1,...
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Quasiconformal map with its dilatation supported in a compact set
Suppose that $\varphi\colon \mathbb{C} \to \mathbb{C}$ is a quasiconformal map (i.e., with $\mu_\varphi < k < 1$ on $\mathbb{C}$) such that its dilatation $\mu_\varphi$ is supported in some ...
- 115
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Find the analytic function $w(z)$
Find the analytic function $w(z)$ that performs a conformal mapping of the region
$$\Omega \equiv \left \{ |z|>1,|z-1|<1 \right \}$$
to the upper half-plane $\text{Im}(w)>0$
My attempt:
First,...
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Quasiconformal homeomorphism with dilatation supported in a strip
Consider a $K$-quasiconformal homeomorphism $\varphi\colon \mathbb{C} \to \mathbb{C}$ such that $\varphi(0) = 0$, $\varphi(1) = 1$, and the corresponding Beltrami coefficient $\mu_{\varphi}$ is ...
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Convergence in Teichmüller space
Question 1
Let $X$ be a compact hyperbolic Riemann surface and let $\{[\mu_n]\}$ be a sequence of points in the Teichmüller space $\mathcal T(X)$.
Further, assume there are maps $f_n:X \xrightarrow{q....
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The set of $K$ quasiconformal mappings between between the disc fixing the origin is compact
Hubbard page 131 gives
Corollary 4.4.3 Denote by $\mathcal F_K(\mathbb D)$ the set of $K$-quasiconformal homeomorphisms $f : \mathbb D \rightarrow \mathbb D$ with $f(0)=0$. Then $\mathcal F_K(\...
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Show that the two elliptic integrals are equal
Prove or disprove: for any $0<a<1$
$$ \int _{-a}^a \frac{du}{\sqrt{(u^2-a^2)(a^2u^2-1)}}=\int_0^\pi \frac{d\theta}{|e^{2i\theta}-a^2|} \tag{1}\label{eq1}$$
Where did I come across these beasts?
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Change of variables in Elliptic equations
Let $\varSigma$ be a domain in $\mathbb{R}^n$, and consider the Laplace equation on $\varSigma$: That is $$ \triangle u =0, $$ in a well-known weak formulation of Elliptic equations (I omit the ...
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A question about Quasi Hyperbolic metric
I am reading Quasicoformally Homogeneous Domains by F. W. Gehring t And B. P. Palks
Let $D$ be a proper subdomain of $\mathbb{R}^n.$ Define the function $\rho.(x) = \frac{1}{dist(x, \partial D)}.$ ...
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Does moving a small enough distance in Teichmüller space change the marking?
Let $S_{g}$ be a genus $g$ closed Riemann surface. The Teichmüller space $\mathcal{T}(S_{g})$ is the set of all pairs $(X,\phi)$ where $X$ is a Riemann surface of genus $g$ and $\phi : S_{g} \...
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Does a homeomorphism on the unit disk, that is the identity on the boundary have bounded displacement?
I want to show a connection between the hyperbolic metric and the boundary values of a homeomorphism. Assuming there is a homeomorphism $f: \mathbb{D} \rightarrow \mathbb{D}$ on the unit disk that can ...
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Homeomorphism but not quasiconformal
Let $f:B^2(0,1)-\bar{B^2}(0,\frac{1}{2}) \to B^2(0,1)-{0}$ be a homeomorphism. To show that $f$ is not quasiconformal.
I can construct one such homeomorphism explicitly. However, I am not getting why ...
user1043248
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Question about Hubbard's analytic definition of quasiconformality. Aren't weak derivatives only defined up to a set of measure zero?
I'm a bit confused about something that appears in the fourth chapter of Hubbard's Teichmüller Theory text. In his statement of Weyl's lemma, he says that if $f$ is a distribution whose weak/...
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Quasiconformal maps take null sets to null sets
Let $f:U\rightarrow V$ be some quasiconformal map in the plane. I want to show that the formula
$$\mathrm{area}(f(E)) = \int_{E}|f_z|^2-|f_{\bar{z}}|^2\mathrm{d}x\, \mathrm{d}y$$
is valid.
There are ...
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$p$-admissible weight, which is not an $A_p$ weight
Let $\Omega\subset\mathbb{R}^N$ be a bounded and smooth domain with $N\geq 2$. Let us consider the following class of weights
$$
B_s=\{w\in W_p: w^{-s}\in L^1(\Omega)\text{ for some }s\in Y\},\quad 1&...
- 421
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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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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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'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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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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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 ...
- 25k
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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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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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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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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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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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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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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 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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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 ...
- 25k
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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 $\...
- 25k
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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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