# Questions tagged [quasiconformal]

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...