Questions tagged [quasiconformal]

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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 ...
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30 views

$L^\infty$ norm of Schwarzian derivative implies univalence?

On page 26 of these notes (Riemann surfaces, dynamics and geometry) there is a theorem 2.13 which says Let $f: \mathbb H \to \hat{\mathbb C}$ be a holomorphic map, satisfying $\|Sf\|_{\infty} <1/2$...
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1 vote
1 answer
149 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(\...
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7 votes
2 answers
158 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 $\sup\{\mu(z):z \in \text{Domain}\{f\}\}<1$ ...
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  • 485
2 votes
1 answer
104 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 ...
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  • 485
2 votes
1 answer
169 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 ...
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  • 485
10 votes
1 answer
661 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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0 answers
114 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 $\mathbb{T}^2$. ...
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  • 6,356
7 votes
1 answer
291 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 ...
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121 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 ...
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2 votes
1 answer
111 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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80 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 ...
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1 answer
593 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 ...
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6 votes
0 answers
234 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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3 votes
0 answers
155 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\...
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0 answers
88 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 $f_1 \colon \mathbb D \to D$ ...
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74 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-...
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4 votes
1 answer
294 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 ...
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4 votes
0 answers
182 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 ...
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  • 189
4 votes
1 answer
310 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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9 votes
0 answers
187 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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1 vote
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56 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}\...
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9 votes
2 answers
214 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 ...
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4 votes
0 answers
75 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 $(\mathbb{D}^n)^o$ ...
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3 votes
0 answers
86 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 $l,h>0$. ...
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3 votes
0 answers
51 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 ...
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1 vote
0 answers
45 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 ...
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13 votes
3 answers
797 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 $d \ge 3$ ...
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5 votes
1 answer
214 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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1 vote
1 answer
86 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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4 votes
1 answer
521 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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4 votes
0 answers
79 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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1 vote
0 answers
91 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 $\...
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3 votes
0 answers
108 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 ...
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  • 1,396
3 votes
1 answer
160 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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  • 179
2 votes
0 answers
91 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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1 vote
0 answers
55 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 ...
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0 votes
1 answer
74 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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1 vote
0 answers
43 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^...
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3 votes
1 answer
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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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0 answers
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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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5 votes
2 answers
278 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 \...
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6 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 ...
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5 votes
1 answer
193 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}} &...
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1 vote
1 answer
170 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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1 answer
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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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  • 101
2 votes
1 answer
264 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 $z \in X$: $...
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2 votes
1 answer
258 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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  • 1,986
4 votes
2 answers
702 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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  • 1,986
8 votes
1 answer
406 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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