Questions tagged [conformal-maps]

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

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11 votes
3 answers
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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
133 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 ...
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2 votes
0 answers
44 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 ...
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6 votes
0 answers
109 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 ...
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1 vote
1 answer
105 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
51 views

On Sobolev's inequality for weakly conformal maps

Suppose $u\in W^{2,p}(B^2,\mathbb{R}^n)$, $1<p<2$, is weakly conformal, that is $$|u_x|=|u_y|,\quad u_x\cdot u_y=0$$ for almost every $(x,y)\in B^2$. Here $B^2$ is the unit open ball in $\mathbb{...
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1 vote
2 answers
395 views

What is a simplified intuitive explanation of conformal invariance? [closed]

Can the concept of conformal map and conformal Invariance be explained in very general terms, preferably in high school/undergrad-level Mathematics? Abstracting away from the applications in physics (...
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0 answers
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Distance Metric on a Polytope

Primary Question: Is it possible to define a distance metric on a polytope (or permutohedron in particular)? I am aware that neither is a smooth, Riemannian manifold; however, computer scientists have ...
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  • 485
0 votes
1 answer
209 views

Existence of an inverse to the Schwarz-Christoffel mapping [closed]

As an elementary result in complex analysis, one can use the argument principle to show that the Schwarz-Christoffel transform is injective on the interior of the polygon to which it maps. Could this ...
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0 votes
1 answer
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Gehring Lemma in dimension 2

In Iwaniec's paper presenting the Gehring Lemma, the embedding used is $W^{1,p}\hookrightarrow L^2$ with $p=\frac{2d}{d+2}$. Question. What about dimension 2: can we actually go down to $p=1$?
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2 votes
0 answers
72 views

Finding $\Omega$ such that the 1-form $\Omega^2 \omega$ is $L^2$ orthogonal to conformal killing vector fields on $S^2$

Consider the space $\mathcal{A}$ of functions $\Omega$ such that $\Omega^2 \gamma_0$ is isometric to the round sphere, where $\gamma_0$ is the round sphere. (so $\Omega^2 \gamma_0$ is of constant ...
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3 votes
1 answer
319 views

Reference on boundary behavior of conformal maps

I am looking for some results on the boundary behavior of conformal maps between simply connected domains. In particular I am interested in conformal maps between $\mathbb{C}-\Delta$, where $\Delta$ ...
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13 votes
1 answer
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How to interpret Gauss's late fragments on conformal mapping of the interior of an ellipse (to the unit disk) in modern mathematical terms?

My question refers to some not very well known (and unpublished) fragments of Gauss that treat the problem of finding a conformal mapping (angle-preserving mapping) in the complex plane from the ...
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19 votes
3 answers
2k views

How to find a conformal map of the unit disk on a given simply-connected domain

By the classical Riemann Theorem, each bounded simply-connected domain in the complex plane is the image of the unit disk under a conformal transformation, which can be illustrated drawing images of ...
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6 votes
1 answer
364 views

Factorization of conformal maps between annuli

Consider two doubly-connected open subsets $A$ and $A'$ of the Riemann sphere. We assume these two domains to be of same modulus (the moduli space being one real parameter), i.e. we assume that there ...
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4 votes
2 answers
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Non-bijective conformal maps between annuli

I need to answer the following question, hopefully in the negative. Question: Does there exist a conformal map $f$ of degree $1$ from the annulus $\{1<|z|<R\}$ to the punctured disk $\{0<|...
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0 votes
1 answer
365 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 ...
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8 votes
1 answer
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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 ...
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11 votes
1 answer
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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 ...
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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?
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