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Questions tagged [quasiconformal]

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Modulus of image of a curve family in a rectangle

I don't expect to get a positive answer to this question but I may as well try. Let $R$ be the rectangle in $\mathbb{C}$ given by $\{z=x+iy: 0\leq x \leq l, 0 \leq y \leq h\}$ for some $l,h>0$. ...
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Extremal metric for image of a curve family

Let $U\subset \mathbb{C}$ be a domain and $\Gamma$ some family of curves in $U$ with $\textrm{mod}(\Gamma)<\infty$ and such that $\rho$ is an extremal metric for the modulus. Suppose we are given a ...
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Explicit Quasisymmetric embedding into Euclidean space

It is known that every doubling metric space admits quasisymmetric map into Euclidean space. My question is, is there a known explicit (closed-form) quasisymmetry from the Heisenberg group into a ...
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3answers
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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 $df^Tdf =(\det df)^{\frac{2}{d}} \, \text{Id}_{TM}$. Assume $d \ge 3$ ...
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1answer
183 views

Regularity of the Jacobian of a $W^{2,n}$ Sobolev mapping

Given a mapping in the Sobolev space $f\in W^{2,n}_{\rm loc}(\mathbb{R}^n,\mathbb{R}^n)$ I would like to know what is the Sobolev regularity of the Jacobian $J_f=\operatorname{det} Df$. It is well ...
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1answer
54 views

Equality on $\partial \mathbb{H}$ of lifts for isotopy to a conformal map

Let $\mathbb{H} \subset \mathbb{C}$ be the upper half plane. First recall the following statement: if $f^* \colon \mathbb{H} \rightarrow \mathbb{H}$ is quasi-conformal (qc), then there exists an ...
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1answer
130 views

Clarification on Beltrami Differentials

I have troubles with the theory of existence of quasi-conformal homeomorphisms realizing Beltrami coefficients. Let $X$ be a (compact) Riemann surface and $f \colon X \rightarrow \mathbb{C}$ be smooth....
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Conformal $L^p$ rigidity of Riemannian manifolds

$\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ $\newcommand{\R}{\mathbb{R}}$ $\newcommand{\CO}[1]{\text{CO}(#1)}$ $\newcommand{\dist}{\operatorname{dist}}$ $\newcommand{\g}{\mathfrak{g}}...
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“Quasiconformal” projections from Heisenberg group to the plane

Let $G$ be the 3-dimensional Heisenberg group equipped with its Carnot-Caratheodory subriemannian metric $d_{G}$. Let $U$ be a domain in $G$ of the form $V \times I$, where $V$ is an open subset of $\...
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Degenerate Beltrami equation and inverse

The Beltrami equation $f_{\bar{z}}=\mu(z)f_{z}$ is degenerate when $\left \| \mu \right \|_{\infty}=1$. For these equations, Lehto and David among others have given conditions for existence. The Lehto ...
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1answer
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Non-injective continuous maps that appear quasiconformal

Suppose that I have a continuous surjection $f: U \rightarrow V$ between two open subsets of the plane. Suppose that $f$ appears to be quasiconformal in the sense that there is a uniform constant $K \...
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Mollifying Green's functions with heat kernel and conformal invariance

For domain D consider Green's fcn $G_{D}(x,y)$ and heat kernel $H_{D}$ and mollify $$K_{D}(x,y,t)=\int_{D}\int_{D}H_{D}(x,w,t)H_{D}(u,y,t)G_{D}(w,u)d^{2}wd^{2}u.$$ The green's fcn satisfies $G_{D}(x,...
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Quasiconformal constant in Nielsen isomorphism theorem

Let $\rho_1$ and $\rho_2$ be two faithfull and discrete representations of the fundamental group of a compact surface into $PSL(2,\mathbb{R})$. The Nielsen isomorphism theorem says that there exists a ...
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1answer
54 views

quasi-conformal embedding of Carnot group into euclidean space

By Pansu's theorem, there are no bi-Lipschitz embeddings of Carnot groups (with exception of the Euclidean space itself) into Euclidean space. Do there exist quasi-conformal embeddings (into Eucl. sp.)...
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quasiconformal groups construction

I have seen the following statement recently: Let $H$ be a Mobius group acting on $\mathbb{S}^n$ and $f$ be a $K$-quasiconformal self-homeomorphism of the $n$-sphere, then the group $fHf^{-1}$ is $K^...
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Assuming admissible functions $\rho$ are continuous in definition of conformal modulus

It's stated in Väisälä's 'Lectures on n-dimensional quasiconformal mappings' (p. 20) that, in the geometric definition of a quasiconformal mapping, that the modulus of a family of curves associated to ...
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conformal deformation with fixed boundaries

For a flat plane with certain boundary, e.g., a rectangular patch, is it possible to conformally displace or deform such patch to a curved bump with exact same boundary? In this thesis, Dr. Keenan ...
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2answers
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Are quasi-Möbius maps always quasi-conformal?

The article "Quasimöbius maps" by Jussi Väisälä states that one always has the implication QM $\implies$ QC. But a proof is only given in for maps of the form $f:\dot{A} \to \dot{Y}$ where $A \subset \...
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What is the Beltrami differential?

Let $R,S$ be Riemann surfaces and $f: R \to S$ an orientation preserving diffeomorphism. Then $f$ determines what is called a Beltrami differential denoted by $\mu \frac{d\bar{z}}{dz}$. Local ...
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$L^p$ stability of the Beltrami equation

Let's assume that $f$ is a quasiconformal homeomorphism of $\mathbb{C}$ with Beltrami coefficient $\mu = \frac{\bar{\partial} f}{\partial f}$. Notice that by definition $\Vert \mu \Vert _{L^{\infty}} &...
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1answer
131 views

Quasiconformal extensions of diffeomorphisms

Let $\gamma:\mathbb R\to\mathbb R$ be an increasing diffeomorphism. Then it is well known that there exist quasiconformal mappings of the upper half plane which extends $\gamma$. One way to construct ...
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1answer
157 views

quasiconformal across the real line

My question is from page 194,line 12 from below, in the book Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane, written by Astala, Iwaniec and Martin. Let $F:\mathbb{C}\...
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1answer
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About a definition of quasi-conformal maps

A book I'm reading gives the following definition for quasi-conformal maps: If $f$ is a homeomorphism of a metric space X to itself, $f$ is K-quasi-conformal if and only if for all $z \in X$: $...
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1answer
244 views

hayman's result for $ A^2(D) $

Consider injective homolomorphic functions $f:\mathbb D\to \mathbb C$ on the unit disk $|z|\leq 1$, normalized by the conditions $f(0)=0$ and $f'(0)=1$. Thus for $|z|\leq 1$ we have $ f(z)=\...
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2answers
561 views

Iwaniec's conjecture

Does anyone know whether there is any geometric applications of the Iwaniec's conjecture on $ l^p $ bound of Beurling Alfhors transform (or the complex Hilbert transform). One application could have ...
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1answer
338 views

quasi conformal, area preserving homomorphisms of the disc

Restricting a quasi-conformal homeomorphism of the disc to the boundary gives a surjective homomorphism from $QC(D^2)$ (quasi-conformal homeos of $D^2$) to $QS(S^1)$ (quasi-symmetric homeos of the ...
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1answer
184 views

Higher dimensional analogue of Kellog's theorem? (Holder continuity of solution to Dirichlet problem with Holder continuous boundary data)

Let $f:S^n\to C$ be a continuous function, $n\geq 1$. When $n=1$, this is a well-known theorem, called Kellog's theorem (or sometimes Kellog-Warschawski's theorem) which states the following Theorem: ...
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1answer
207 views

Analytic curve on Riemann surface

Suppose there is a closed analytic curve $C$ on a Riemann surface $S$, that is the image of a map $\gamma$ from the equator $E$ of the Riemann sphere to the surface $S$ which is a restriction of a ...
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1answer
251 views

Teichmuller Theory question : Beltrami forms on hyperbolic Riemann surfaces whose lifts are smooth upto the boundary of $\mathbb{D}$

Hello, my question is related to Teichmuller Theory. Let $D$ be the open unit disk and $X=D/{\Gamma}$ be a hyperbolic Riemann surface of the Fuchsian group $\Gamma$. In Teichmuller theory, we have ...
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1answer
247 views

Boundary regularity of quasiconformal homeomorphisms of the unit disk ?

Hello, I asked this question before, but didn't get any response, so I took the liberty of asking once again , with slightly modified version of the question: Consider an orientation-preserving ...
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1answer
310 views

A Fact Of Quasiconformal Map

We just consider puntured unit disk $\triangle^{*}$ in $\mathbb{C}$. $f$ is a bounded quasiconformal map on $\triangle^{*}$. Why $f$ can extend to the origin,becoming quasiconformal map on the whole ...
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410 views

Boundary regularity of the solution to the Beltrami equation

Hello, this question might sound a little vague, but I still dare to state , and I am basically requesting for some reference: Let us consider the orientation-preserving homeomorphic solutions $f: D \...
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1answer
299 views

Is the disk quasiconformally isomorphic to the plane?

(This question might turn out to be too elementary for this site, if so I'm sorry, but I can't find the answer anywhere.) Does there exist a function $\; f : \{z\in \mathbb{C} : |z| < 1\} \to \...
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1answer
438 views

A regularity question on the Beltrami equation $ f_\bar{z} =\mu . f_z$ on $D$

Hello, This question is related to Chapter V, lemma 3 on page 54 of Lars Ahlfors' 'Lectures on Quasiconformal mappings' which states : If $\mu:\mathbb{C}\to \mathbb{D} \in W^{1,p}(\mathbb{C}), p ...
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1answer
256 views

Ahlfors' proof of Locally K-Quasiconformal to K-Quasiconformal

This is a question I originally posted in Math Stack Exchange, but perhaps the question was too specialized, so I thought I'd post it here instead I'm currently reading through "Lectures on ...
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1answer
435 views

Two questions from Hubbard's Teichmuller theory book Vol I, P. 130 , Thm 4.4.1, ( QC maps )

I was studying Theorem 4.4.1 from John H. Hubbard's Teichmuller Theory, vol I, Theorem 4.4.1 ( P. 129 ) which states : Let $X,Y$ be two hyperbolic Riemann surfaces with hyperbolic metrics $d_X,d_Y$ ...
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1answer
660 views

A quick and elementary question from Hubbard's Teichmuller Theory : Volume I

Hi, On page 120, chapter 4, proposition 4.2.7 in Hubbard's Teichmuller Theory book, volume 1, he proves : Let $U,V$ be open in $C, f:U \to V $ be a homeomorphism and the restriction of $f$ on $U \...
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3answers
542 views

How to prove/disprove that quasiconformal maps send measure-zero sets to measure-zero sets

$Qn#1 $ : Let $f:U\to V$ be a $K$ quasiconformal homeomorphism ( NOT diffeomorphism ) of plane open subsets of $C$. By my definition of quasiconformality, I mean 1)$f$ is continuous, 2)the weak ...
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1answer
183 views

Coefficients of lacunary series on quasiconformally transformed unit disk

Say I have a lacunary $q$ series $s(q)=\sum_{n=0}^{\infty} a_{n}q^{n}$ , and I have a quasiconformal transformation $\xi$ which preserves the boundary of the unit disk in $\mathbb{C}$ such that if $|q|...