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0answers
24 views

Analytical form of conformal mapping of simple closed curve

I want to create a mapping to map a simple closed curve to a unit disk, in analytic forms. The curve is simple enough, with several segments: each with an analytical form, but no general analytical ...
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1answer
70 views

Existence of certain analytic functions

Let $D_1,D_2\subset\mathbb{C}$ be two disjoint disks. Is there an analytic function defined on the upper half plane $f:\mathbb{H}\rightarrow\mathbb{C}$ such that $f$ takes each value in $D_1$ exactly ...
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0answers
34 views

comformal surface parameterization of simply-connected Riemann surface with boundary

$M$ and $S$ are two simply-connected surfaces with boundaries $\partial M =\partial S \neq \emptyset$, respectively. Both $M$ and $S$ are disk-like and smooth enough. Here, we assume $M$ is planar ...
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1answer
131 views

Quasiconformal map between the complex plane and a disk

According to the Poincaré-Koebe theorem, it is known that the unit disk $\mathbb D$ and the complex plane $\mathbb C$ aren't conformally equivalent. My question is maybe naive, but I was wondering if ...
2
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1answer
111 views

Lift of a homeomorphism $f$ between two (hyperbolic) surfaces $X,Y$

Let $X,Y$ be two hyperbolic Riemann surfaces (i.e. they have universal cover the upper half plane $\mathbb{H}$). Let $\pi_X:\mathbb{H}\to X, \pi_Y:\mathbb{H}\to Y $ be the corresponding covering maps. ...
2
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1answer
45 views

Give a example about invariant ergodic measure and quasi-symmetric mapping

Is there a example $(X,f,\mu)$ such that $X$ is a closed subset of Euclidean space, $f$ be a quasi-symmetric mapping but not a Lipschitz mapping, $f(X)=X$, $\mu$ is a finite measure on $X$ that is ...
2
votes
1answer
415 views

Precise definition of conformal structure based on a Riemannian metric on a Riemann surface

As I read the literature, I keep having some doubt about what a " conformal structure on a Riemann surface " exactly means. ( You can assume all the Riemann surface in this literature have universal ...
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2answers
382 views

quasiconformal “automorphism” groups of julia sets

To motivate this question, let me begin with a picture: Each letter labels a "blob" of this quartic julia set. (is there a technical term for these parts?). Because of resolution limitations I ...