# Questions tagged [conformal-maps]

For conformal mappings of Riemann surfaces (for example, domains in the complex plane).

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4 votes
1 answer
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### Conformal maps between two given domains

Consider two domains \begin{aligned} D_1&=\{x=(x_1,x_2,...,x_n)\in\mathbb{R}^n:x_n\leq 0\},\\ D_2&=\{x=(x_1,x_2,...,x_n)\in\mathbb{R}^n:x_n\leq \psi(x_1,x_2,...,x_{n-1})\}, \end{aligned} ...
• 1,049
1 vote
0 answers
36 views

### Splitting of the conformal group into $PSL(2,\mathbb{R})$ and other factorizations

In 1+1 dimensions of Minkowski spacetime, the conformal group can be split into two copies of $PSL(2,\mathbb{R})$ acting on null lines. I'm curious to know if a similar split exists for the conformal ...
4 votes
1 answer
164 views

### Conformal map between flat and hyperbolic torus with a boundary

I am confused because I can define two very different complex structures on the torus with a puncture/boundary. For my first construction, I can imagine removing a disk from a flat torus, inheriting ...
• 173
3 votes
1 answer
131 views

### Do we have uniformization theorems for fractional dimensional spaces?

The Riemann mapping theorem in $\mathbb{R}^2$ is known not to generalize well in higher dimensions and is basically trivial in lower dimensions. I’m interested in how it generalizes for fractional ...
• 2,018
4 votes
1 answer
254 views

### Riemann mapping theorem with smooth boundary

This is closely related to the question here. The setup is that $U\subset\mathbb{C}$ is an open bounded simply connected domain with $C^\infty$ boundary. If $\phi:U\rightarrow\mathbb{D}$ is a ...
• 337
0 votes
0 answers
122 views

### Conformal diffeomorphism of $\mathbb R^k$

Let $f$ be a conformal diffeomorphism between $\mathbb{R}^k$, where $k\geq 2$, and its Euclidean metric. It follows from complex analysis and Liouville's theorem that $f$ can only be affine. Now, ...
• 305
1 vote
0 answers
122 views

### Numerically compute the Schwarz-Christoffel mapping to the square

I want to map the upper-half plane $$\mathbb H:=\{z\in\mathbb C:\Im(z)>0\}$$ to $[0,1)^2$ by a conformal map. If I got this right, then such a mapping is given by the Schwarz-Christoffel mapping to ...
0 votes
0 answers
45 views

### How do we solve this rather simple ODE (Loewner equation with driving function $\sqrt t$)?

Remember the following result for the Loewner equation: If $\lambda:[0,\infty)\to\mathbb R$ is continuous, then for all $z\in\mathbb C\setminus\{\lambda(0)\}$ there is a uniqe $\zeta(z)\in(0,\infty]$ ...
3 votes
0 answers
62 views

### Boundary behavior of conformal map on domain satisfying an exterior sphere condition

I'm in the middle of a project concerning a Bernoulli-type free boundary problem in $\mathbb{R}^2$ and, as part of this project, I would like to understand the boundary behavior of conformal maps on ...
• 603
11 votes
3 answers
714 views

### Explicit triples of isomorphic Riemann surfaces

Inspired by a discussion with Neil Strickland I am very interested to hear of explicit examples (one per answer, please), as follows. A compact Riemann surface can be presented in many different ways....
2 votes
1 answer
216 views

### Explicit universal covering map for higher genus algebraic curves

Suppose I have a projective plane curve $C = V(F)$ defined over $\mathbb{C}$, where $F$ is some homogeneous polynomial in three variables. For the sake of simplicity, let's assume that $C$ is ...
• 123
2 votes
0 answers
54 views

### Triangulations with discrete metrics and conformal equivalence

A discrete metric for a triangulation of a 2-dimensional manifold is a map associating $\mathbb{R}_+$-valued lengths to all edges, such that the triangle inequality holds on every triangle. In many ...
• 2,799
7 votes
0 answers
142 views

### What are the generators and relations of the conformal cobordism category?

According to a definition by Segal, a $2$-dimensional CFT is a symmetric monoidal functor from the category of oriented conformal cobordisms to the cateogry of projective complex vectorspaces. Coming ...
• 2,799
1 vote
1 answer
113 views

### Is there a non-trivial, exact analytic (symbolic) conformal map from some polygon to some rectangle?

I'm looking for any example of a conformal map $m: P \to Q$ where $P$ is some polygon of at least $4$ sides, $Q$ is a rectangle, and and $m$ is not linear (so $P$ and $Q$ are not merely scaled, ...
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3 votes
0 answers
53 views

• 1,754
0 votes
1 answer
390 views

### Riemann mapping

Let in the complex plane be a bounded Jordan region T (that is a bounded and simply connected set with the boundary a Jordan curve), containing the origin, with its Riemann mapping onto the open unit ...
• 71
8 votes
1 answer
2k views

### Is there a manifold structure on a space of conformal maps?

I would be very grateful for any information or pointers for the following: 1) Fix an open subset $U$ of $\mathbb{CP}^1$. a) Does the set of all holomorphic maps from $U$ to $\mathbb{C}$ (with the ...
• 81
11 votes
1 answer
5k views

### Conformal maps of doubly connected regions to annuli.

In another question here on MO, Anweshi asks if any doubly connected region in the complex plane can be conformally mapped to some annulus. The answer to this is yes. But the fact is that two annuli ...
• 2,030
7 votes
3 answers
2k views

### Riemann mapping for doubly connected regions

Remove the closure of simply connected region from the interior of a simply connected region. Is it true that the resulting domain can be mapped conformally to some annulus?
• 7,212