Questions tagged [quasiconformal]

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2answers
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
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1answer
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
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1answer
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
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1answer
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}}...
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0answers
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 ...
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0answers
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
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0answers
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
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1answer
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
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0answers
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
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1answer
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 ...
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0answers
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
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1answer
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
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0answers
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, ...
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0answers
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
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0answers
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
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0answers
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
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1answer
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
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0answers
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
1answer
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 $C^\infty$ ...
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0answers
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 ...
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0answers
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
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0answers
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
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0answers
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
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0answers
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
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0answers
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
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0answers
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
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3answers
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
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1answer
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
1answer
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
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1answer
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
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0answers
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}}...
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0answers
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
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0answers
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
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1answer
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 \...
2
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0answers
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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0answers
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
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1answer
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.)...
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0answers
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
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1answer
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 ...
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0answers
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
2answers
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
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0answers
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
1answer
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
1answer
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
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1answer
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
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1answer
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
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1answer
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
2answers
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
1answer
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
2answers
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: ...