Questions tagged [quasiconformal]
The quasiconformal tag has no usage guidance.
66
questions
13
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3
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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 $d \ge 3$ ...
10
votes
1
answer
689
views
How to shrink a square with minimal distortion?
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10
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0
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Bi-Lipschitz mappings
Assume that we have a bi-Lipschitz mapping $f:\bar{\mathbb{B}}^n(0,1)\to\mathbb{R}^n$. The mapping need not be smooth anywhere and it may happen that it cannot be extended to a homeomorphism of a ...
9
votes
2
answers
278
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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 ...
9
votes
0
answers
203
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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 ...
8
votes
2
answers
270
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 $\sup\{\mu(z):z \in \text{Domain}\{f\}\}<1$ ...
8
votes
1
answer
457
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 ...
7
votes
1
answer
384
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 ...
6
votes
1
answer
264
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 ...
6
votes
0
answers
244
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Do asymptotically conformal maps converge to a weakly conformal map?
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Let $\M,\N$ be two-dimensional smooth, ...
6
votes
0
answers
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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 ...
5
votes
1
answer
819
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....
5
votes
1
answer
209
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}} &...
5
votes
2
answers
343
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
130
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 ...
4
votes
1
answer
834
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 ...
4
votes
1
answer
442
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
1
answer
718
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 \...
4
votes
2
answers
773
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 ...
4
votes
1
answer
335
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 \...
4
votes
1
answer
481
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 $C^\infty$ ...
4
votes
1
answer
579
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 ...
4
votes
0
answers
283
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
0
answers
83
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 $(\mathbb{D}^n)^o$ ...
4
votes
0
answers
83
views
Conformal $L^p$ rigidity of Riemannian manifolds
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3
votes
1
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187
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 \...
3
votes
1
answer
111
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 ...
3
votes
0
answers
82
views
Conformal welding and Jordan loop consequences?
In the similar context as 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 $D$ ...
3
votes
0
answers
158
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\...
3
votes
0
answers
82
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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-...
3
votes
0
answers
109
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
53
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 ...
3
votes
0
answers
114
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
0
answers
427
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 \...
2
votes
3
answers
728
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 ...
2
votes
1
answer
311
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
132
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 ...
2
votes
1
answer
111
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
261
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)=\...
2
votes
1
answer
143
views
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} \...
2
votes
1
answer
324
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 ...
2
votes
1
answer
193
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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|...
2
votes
1
answer
229
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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 ...
2
votes
0
answers
71
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Equicontinuity, Beltrami coefficients and Sequence of top/bottom semi-annuli
In the Beltrami equation literature, one approach to showing equicontinuity for pairs $(f_{n},\mu_{n})$ (where $\partial_{\bar{z}}f_{n}=\mu_{n}(z)\partial_{z}f_{n}$) is via the relations of moduli and ...
2
votes
0
answers
108
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 $f_1 \colon \mathbb D \to D$ ...
2
votes
1
answer
107
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 ...
2
votes
0
answers
103
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,...
1
vote
1
answer
298
views
Motivation for the definition of $L^p$ norm for quadratic and Beltrami differentials
According to Riemann surfaces, dynamics and
geometry
by C. McMullen (Course notes), the definition for a quadratic differential $\phi$ on a Riemann surface $X$ is given by
$$
\|\phi\|_p = \left(\...
1
vote
2
answers
382
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 ...
1
vote
1
answer
197
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 ...