All Questions
Tagged with cv.complex-variables conformal-maps
17 questions
24
votes
3
answers
3k
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 ...
- 41.8k
15
votes
1
answer
2k
views
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My question refers to some not very well known (and unpublished) fragments of Gauss that treat the problem of finding a conformal mapping (angle-preserving mapping) in the complex plane from the ...
- 2,099
12
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 ...
- 2,050
11
votes
3
answers
748
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....
8
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?
- 7,442
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
6
votes
1
answer
467
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 ...
4
votes
2
answers
2k
views
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<|...
- 1,804
4
votes
1
answer
451
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 ...
- 439
3
votes
1
answer
590
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 $\Delta$ ...
3
votes
0
answers
114
views
Conformal welding and Jordan loop consequences?
In the similar context as 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 $D$ ...
- 5,474
3
votes
0
answers
90
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 ...
- 683
2
votes
2
answers
2k
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 (...
- 157
2
votes
0
answers
48
views
Convergence of finite-difference method for Cauchy-Riemann equations
Let $I\subseteq \mathbb{R}$ an open interval. Let $f:I\rightarrow \mathbb{C}$ real analytic. Suppose we want to numerically compute an analytic extension of $f$.
We will assume the following: we are ...
- 312
0
votes
1
answer
405
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
0
votes
1
answer
575
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 ...
- 547
0
votes
0
answers
66
views
Uniformization and constructive analytic continuation of Taylor-Maclaurin series
Context. In their paper, "Uniformization and Constructive Analytic Continuation of Taylor Series", Costin and Dunne present a constructive method to greatly increase the accuracy of a ...
