2
votes
1answer
91 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 ...
-2
votes
1answer
129 views

Holomorphic maps on $\mathbb{R}^{n}$ (for n not necessarily even)

Edit according to the comment of user36931 I remove the "motivation" from the previous version and I add an statement to the first question We consider the following two classes of smooth maps on ...
2
votes
1answer
154 views

Why do holomorphic maps increase extremal length?

Let $\mathcal{L}$ denote extremal length. The following theorem appears in http://arxiv.org/abs/math/0505191. Theorem Let $U$ and $V$ be Riemann surfaces, and let $f:U\rightarrow V$ be a holomorphic ...
2
votes
1answer
212 views

hayman's result for $ A^2(D) $

Consider injective homolomorphic functions $f:\mathbb D\to \mathbb C$ on the unit disk $|z|\leq 1$, normalized by the conditions $f(0)=0$ and $f'(0)=1$. Thus for $|z|\leq 1$ we have $ ...
0
votes
1answer
134 views

Inverse “Riemann mapping” [closed]

The Riemann mapping theorem states, that any simply connected domain $U \subset \mathbb C$ can be conformally mapped to the open unit disk $D$. I.e. there is a Diffeomorphism $\Psi: D \to U$ such that ...
1
vote
1answer
203 views

Two different forms of Schwarz-Christoffel-Mapping of unit disk to rectangle. Are they identical?

I found two different equations for the Schwarz-Christoffel-mapping of a unit disk to a rectangle (which are the general form of the SC-mapping, I guess). The first, e.g. from Link, page 20, is ...
1
vote
1answer
288 views

How to determine bounds on the extremal length around annuli?

I wish to determine bounds for the sum of moduli of a family of topological annuli in the complex plane. Towards that end I would like to ask a question about the closely related concept of extremal ...
5
votes
1answer
225 views

Adding segments to an annulus - a question regarding the conformal modulus.

Let $A \subset \mathbb{C}$ be an topological annulus, i.e. a region of $\mathbb{C}$ bounded by two disjoint Jordan curves. Let $B \subset \mathbb{C}$ be a quadrilateral, i.e. a topological disc with ...
2
votes
0answers
283 views

Collection of charged line segments in 2D - where do electric field lines meet it?

Suppose we have a collection of charged line segments in 2D. I'd like to be able to do two things : from an arbitrary point in the plane, follow the electric field and find where it meets the ...
7
votes
1answer
1k 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 ...
0
votes
0answers
238 views

Errata in “Conformal Mapping: Method and Applications” from Schinzinger

hi, i'm studying the book "Conformal Mapping: Method and Applications" from Schinzinger and more precisely the chapter 6 concerning non-planar field. In section 6.1.3 the author proposes to solve a ...
0
votes
1answer
446 views

Invariance of the cylindrical Laplace equation under conformal transform

hello, it is often said that a conformal mapping preserves the Laplace equetion in 2D. However, if this is true for the cartesian coordinates (x,y), where the laplacian is: $$ \frac{\partial^2 ...