# Questions tagged [quasiconformal]

The quasiconformal tag has no usage guidance.

62
questions

**5**

votes

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137 views

### Quasiconformal maps in arbitrary dimensions

I am aware that a quasiconformal map satifies the formula
$$
\frac{\partial f}{\partial \overline{z}} = \mu(z) \frac{\partial f}{\partial z}
$$
where $\sup\{\mu(z):z \in \text{Domain}\{f\}\}<1$ ...

**2**

votes

**1**answer

95 views

### Quasiconformal map from a subset of $\mathbb{C}$ to a polytope

Question. Does a quasiconformal map exist between a subset of $\mathbb{C}$ (such
as a unit disc or rectangle) and a polytope?
Here, I take a polytope to be a two-dimensional surface that could be ...

**2**

votes

**1**answer

119 views

### Bicomplex Conjugate Derivative

I have decided to first ask my question and second provide a list of steps I have already considered.
Question: After reading Luna-Elizarrarás, Shapiro, Struppa, and Vajiac - Bicomplex numbers and ...

**8**

votes

**1**answer

517 views

### How to shrink a square with minimal distortion?

$\newcommand{\CO}{\text{CO}_2}$
$\newcommand{\euc}{\mathfrak{e}}$
$\newcommand{\SO}{\text{SO}_2}$
$\newcommand{\al}{\alpha}$
$\newcommand{\dist}{\text{dist}}$
$\newcommand{\Lip}{\text{Lip}_{\text{inj}}...

**0**

votes

**0**answers

36 views

### Harmonic extension and quasiconformal extension

We assume $h:S^1\to S^1$ is a quasisymmetric homeomorphism on the unit circle $S^1$. Let $H$ be the harmonic extension of $h$ in the unit disc, $g$ be a quasiconformal extension of $h$ in the unit ...

**0**

votes

**0**answers

70 views

### Can anyone prove that an IFS cubic Julia Mandelbrot set is not a “real” Mandelbrot set?

This question is the "null question": my proof that these Julia sets are three attraction set Mandelbrot sets is two fold:
The Point at Infinity sets of cubic Julia Mandelbrot sets are ...

**0**

votes

**0**answers

106 views

### Metric obstructions for area-preserving diffeomorphisms with constant singular values

Let $\mathbb{T}^2$ be the topological $2$-dimensional torus, and let $0<\sigma_1 < \sigma_2$ satisfy $\sigma_1 \sigma_2=1$. Let $g$ be an arbitrary smooth Riemannian metric on $\mathbb{T}^2$.
...

**7**

votes

**1**answer

246 views

### A diffeomorphism of the torus with constant singular values

Let $\mathbb{T}^2=\mathbb{S}^1 \times \mathbb{S}^1$ be the flat $2$-dimensional torus, and let $0<\sigma_1 < \sigma_2$ satisfy $\sigma_1 \sigma_2=1$.
Does there exist an area-preserving ...

**5**

votes

**0**answers

120 views

### Is Sobolev limit of bijective maps surjective?

This is a cross-post.
Let $\Omega_1,\Omega_2 \subseteq \mathbb R^2$ be open, connected, bounded, with non-empty $C^1$ boundaries. Let $f_n:\bar\Omega_1 \to \bar\Omega_2$ be $C^1$ be bijective maps ...

**2**

votes

**1**answer

55 views

### The Beltrami equation and Neumann series

Let $\mu: \mathbb{C} \to D(0,1).$ A quasiconformal map is a $W^{1}_{2,loc}-$solution to the Beltrami equation $\bar{\partial}f = \mu \partial f$. In this paper, the authors remark that one can ...

**1**

vote

**0**answers

69 views

### Continuity of conformal grafting, wrt the (infinite type) surface

Say I have two closed hyperbolic surfaces $X,Y$ and a smooth, $(1+\epsilon)$-bilipschitz map $f : X \to Y$ for some small $\epsilon$. Pick a simple closed curve $c \subset X$, and let $X',Y'$ be the ...

**4**

votes

**1**answer

446 views

### The (measurable) Riemann mapping theorem

The Riemann mapping theorem says that a strict, nonempty open subset of the complex plane is conformally equivalent to the unit disk.
The measurable Riemann mapping theorem asserts the existence and ...

**7**

votes

**0**answers

230 views

### Do asymptotically conformal maps converge to a weakly conformal map?

$\newcommand{\CO}{\text{CO}_2}$
$\newcommand{\SO}{\text{SO}_2}$
$\newcommand{\dist}{\text{dist}}$
$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
Let $\M,\N$ be two-dimensional smooth, ...

**3**

votes

**0**answers

153 views

### Convergence of Fuchsian groups and existence of suitable homeomorphisms

Let $(\Gamma_n)_n$ ($\subset PSL(2,\mathbb{R})$) be a sequence of discrete groups, if we say that $(\Gamma_n)_n$ converges to a group $\Gamma$ this means that there exist isomorphisms $\tau_n:\Gamma\...

**2**

votes

**0**answers

86 views

### Conformal welding of rectifiable curves

In classical conformal welding theory, we start with a homeomorphism $h$ of the unit circle and try to find a Jordan domain $D$ together with two conformal isomorphisms $f_1 \colon \mathbb D \to D$ ...

**3**

votes

**0**answers

68 views

### How does one prove that the Teichmuller space of a closed Riemann surface of genus $\geq2$ is uniquely geodesic?

I am reading Masur's paper On a class of geodesic in Teichmuller space. He mentions that $T(S_0)$ where $S_0$ is a closed Riemann surface $g\geq2$ is straight, i.e. uniquely geodesic. It seems a well-...

**4**

votes

**1**answer

211 views

### Curvature estimate for minimal surfaces

I am a bit confused about Theorem 2.16 in the book "A Course in Minimal Surfaces" by Colding and Minicozzi. The authors write that Theorem 2.16 was proved in this paper by Schoen and Simon in the more ...

**4**

votes

**0**answers

132 views

### Bers' simultaneous uniformization

I have been trying to understand Bers' famous paper "Simultaneous Uniformization". Regarding this paper I have a few questions. Any kind of help will be appreciated.
Let $S$ and $S^{'}$ be two ...

**4**

votes

**1**answer

260 views

### Degenerate Beltrami equation

Question:
Let $\mu:\mathbb C\to \mathbb C$ be a $C^\infty$ function satisfying $|\mu|\le 1$.
Let us furthermore assume that the function $\mu$ never takes the value $-1$.
Does there exist a $C^\infty$ ...

**9**

votes

**0**answers

168 views

### A geometric characterization of quasicircles

I'm reading an article by complex analysists.
A Jordan curve $J$ in the extended complex plane $\hat{\mathbb{C}}=\mathbb{C} \cup \{\infty\}$ is called a quasicircle if there is a quasiconformal map ...

**1**

vote

**0**answers

51 views

### Modulus estimate with intersecting annuli (quasi-additivity)

In general for annulus $A\subset \mathbb{C}$ if $A_{1},A_{2}....\subset A$ are disjoint annuli inside it, then we have
$$mod(A)=\frac{1}{2\pi}\int_{A}\int_{A} \frac{1}{|z|^{2}}dz>\frac{1}{2\pi}\...

**4**

votes

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61 views

### Converse to Wolpert's Lemma

Recall Wolpert's lemma:
Let X,Y be hyperbolic surfaces and $f:X\to Y$ a $K$-quasiconformal homeomorphism. For any homotopy class of curves $c$ let $\ell(c)$ denote the length of the geodesic in the ...

**4**

votes

**0**answers

67 views

### Can we approximate harmonic maps which are a.e. orientation-preserving with maps which preserve orientation globally?

Let $\mathbb{D}^n$ be the closed unit ball, and let $f:\mathbb{D}^n \to \mathbb{R}^n$ be harmonic; More precisely, I assume that $f$ is real-analytic and harmonic on the interior $(\mathbb{D}^n)^o$ ...

**3**

votes

**0**answers

73 views

### 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 $l,h>0$. ...

**3**

votes

**0**answers

49 views

### 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 ...

**1**

vote

**0**answers

42 views

### 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 ...

**13**

votes

**3**answers

690 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 $df^Tdf =(\det df)^{\frac{2}{d}} \, \text{Id}_{TM}$.
Assume $d \ge 3$ ...

**5**

votes

**1**answer

203 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 ...

**1**

vote

**1**answer

72 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 ...

**4**

votes

**1**answer

358 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....

**4**

votes

**0**answers

76 views

### 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}}...

**1**

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88 views

### “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 $\...

**3**

votes

**0**answers

105 views

### 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 ...

**3**

votes

**1**answer

124 views

### 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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votes

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84 views

### 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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vote

**0**answers

50 views

### 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 ...

**0**

votes

**1**answer

72 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.)...

**1**

vote

**0**answers

42 views

### 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^...

**3**

votes

**1**answer

103 views

### 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 ...

**0**

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**0**answers

92 views

### 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 ...

**5**

votes

**2**answers

263 views

### 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 \...

**5**

votes

**0**answers

2k views

### 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 ...

**5**

votes

**1**answer

177 views

### $L^p$ 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}} &...

**1**

vote

**1**answer

153 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 ...

**0**

votes

**1**answer

165 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}\...

**2**

votes

**1**answer

228 views

### 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 $z \in X$:
$...

**2**

votes

**1**answer

253 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)=\...

**4**

votes

**2**answers

660 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 ...

**8**

votes

**1**answer

383 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 ...

**0**

votes

**2**answers

242 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: ...