18
votes
Accepted
The (measurable) Riemann mapping theorem
You misstated Riemann's (original) theorem:
a crucial assumption is that your open subset
is simply connected.
Both theorems can be considered as classification theorems
of Riemann surfaces. The ...
- 91.8k
7
votes
Accepted
A diffeomorphism of the torus with constant singular values
There are no non-affine examples with $f$ smooth (or even $C^3$). This follows from the fact that such an $f:\mathbb{T}^2\to\mathbb{T}^2$ would lift to a smooth (local) diffeomorphism $F:\mathbb{R}^2\...
- 108k
7
votes
Accepted
Curvature estimate for minimal surfaces
Theorem 2.16 (for embedded minimal disks) is implicit in Schoen-Simon. There are two steps. The first is that some area (or total curvature) bound on an extrinsic ball implies small total curvature ...
- 86
6
votes
Converse to Wolpert's Lemma
The answer is no, but it's actually a deep question and leads to another metric on the Teichmüller space of surfaces. The minimum quasi-conformal constant of a map in a given homotopy class is the ...
- 10.1k
6
votes
Accepted
Clarification on Beltrami Differentials
I am not really familiar with Imayoshi and Taniguchi's book on Teichmüller theory, but here is my understanding of Beltrami differential, which I learned from Hubbard's book Theichmüller Theory and ...
- 298
6
votes
Accepted
Motivation for the definition of $L^p$ norm for quadratic and Beltrami differentials
The powers of $\rho$ are necessary to make the integrals well-defined. In probably excessive detail:
Because $X$ is a Riemann surface, the integrands here need to be $(1,1)$-forms. This is just the ...
- 76
5
votes
Accepted
Quasiconformal maps in arbitrary dimensions
The correct definition of higher-dimensional quasiconformal maps does not use complex variables. The correct condition is
$$
|Df(x)|\le K |J_f(x)|
$$
where $J_f$ is the Jacobian determinant. An ...
- 12.2k
5
votes
Accepted
How to shrink a square with minimal distortion?
Update. I'll show in the end of this post in the proof that the answer to the question is positive for all $\sigma_1\in (0,1)$.
The square is denoted by $\square$.
Set up. Let us fix $x,y\in (0,1)$ ...
- 28.9k
5
votes
A conformal map whose Jacobian vanishes at a point is constant?
I am not at all an expert in this field. However, I believe I can show that there are no examples where $Df$ vanishes to finite order. Thus, in particular there are no analytic solutions and, if the ...
- 156k
5
votes
A conformal map whose Jacobian vanishes at a point is constant?
I am no expert in this field, but I just looked into Harmonic Morphisms Between Riemannian Manifolds by Paul Baird and John C. Wood, specifically into chapter 14. Here they give more general ...
- 8,597
5
votes
Accepted
Non-injective continuous maps that appear quasiconformal
Such maps are called quasiregular. There is a highly developed theory of them.
Most of the classical theory you can find in the books of Yu. Reshetnyak, Space mappings with bounded distortion, AMS, ...
- 91.8k
5
votes
The Beltrami equation and Neumann series
The story of the solution of the Betrami equation using the Beurling transformation in $L^2$ has an elegant and elementary explanation in:
Adrien Douady and X. Buff, Le théorème d’intégrabilité des ...
- 26.3k
4
votes
Accepted
Quasiconformal map from a subset of $\mathbb{C}$ to a polytope
Yes, there exists a quasiconformal map (even a homeomorphism) from the Riemann sphere
to a polytope. Embedding to $R^3$ is irrelevant here, all we need is the intrinsic
metric, which is a flat metric ...
- 91.8k
3
votes
Converse to Wolpert's Lemma
No. The issue is that quasiconformality constants are much more sensitive to local distortion than lengths. This is morally a $L^\infty$ (quasiconformal constants) to $L^1$ (lengths) comparison: $L^\...
- 680
3
votes
Accepted
quasi-conformal embedding of Carnot group into euclidean space
It depends precisely what you mean by ``quasiconformal embedding.'' There are different definitions that do not agree in complete generality on all metric spaces. (See http://www.ams.org/notices/...
- 46
2
votes
Does moving a small enough distance in Teichmüller space change the marking?
Unless I misunderstand your question, you can take $r = \infty$. This is because we can choose representatives of marked conformal classes to be "Riemann surface structures" on $S_g$. ...
- 28.1k
2
votes
Quasiconformal maps in arbitrary dimensions
The significance of the Beltrami equation $f_{\overline{z}}=\mu f_z$ lies in the fact that it is solvable: for a given $\mu$, $\|\mu\|_\infty<1$ one can find $f$.
For real analytic $\mu$ local ...
- 91.8k
2
votes
Accepted
Regularity of the Jacobian of a $W^{2,n}$ Sobolev mapping
Here is an answer from
Andrea Cianchi:
A form of Hölder's inequality in Orlicz spaces asserts that, if $f_1\in L^{A_1},\ldots,f_n\in L^{A_n}$, and $B$ is such that
$$
A_1^{-1}(t)\cdots A_n^{-1}(t)\leq ...
- 28k
2
votes
Ahlfors' proof of locally K-quasiconformal to K-quasiconformal
If you are still interested in the this problem and in how to make this machinery workS, my suggestion is to taking a carefully reading in the notes by Chris Bishop available in
https://www.math....
2
votes
Ahlfors' proof of locally K-quasiconformal to K-quasiconformal
I can perhaps imagine roughly what the connection to Teichmüller's extremal problem would be, but I'm having enough difficulty unpacking both Ahlfor's proof and Shishikura's remark that I don't think ...
- 565
2
votes
A conformal map whose Jacobian vanishes at a point is constant?
In a different direction, there are solutions where $f$ is $C^1$ and the metric $g_M$ is $C^0$. Let $\phi : \mathbb{R}_{>0} \to \mathbb{R}_{>0}$ be a $C^1$ function such that $\lim_{s \to 0^+} \...
- 156k
1
vote
Accepted
Bicomplex Conjugate Derivative
Since $F$ is bicomplex-holomorphic, we necessarily have $\frac{\partial F}{\partial Z^\dagger} = 0$, and your condition, let's call it the bicomplex Beltrami equation,
$$
\frac{\partial F}{\partial Z^\...
- 7,127
1
vote
Degenerate Beltrami equation
A reference for these types of equations is in Astala, Iwaniec and Martin chapter 20. One of the main difficulties is showing that $f$ is a homeomorphism. In general it will not be the case. For ...
- 11
1
vote
A regularity question on the Beltrami equation $ f_\bar{z} =\mu . f_z$ on $D$
The most comprehensive source for the Beltrami equation in the plane is the book listed below and you should search it for the answers to your questions.
K.Astala, T.Iwaniec, G.Martin,
Elliptic ...
- 28k
Only top scored, non community-wiki answers of a minimum length are eligible