1
vote
1answer
81 views

How far do conjugated Mobius transforms move points?

I start with an automorphism $f$ of the complex unit disc $S^1 = \{ z \in \mathbb{C} : |z| \leq 1\}$. I assume that such a map is given by a Mobius transform, namely $$ f(z) = \frac{z - ...
-2
votes
1answer
127 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 ...
1
vote
1answer
198 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 ...
2
votes
1answer
285 views

Variant of the Riemann Mapping Theorem for $Conf(\mathbb H^2)$?

According to the Riemann mapping theorem it is possible to map a simply connected open subset $B \subset \mathbb C$ into any other $B' \subset \mathbb C$ by a (bi-)holomorphic mapping. Moreover, such ...
1
vote
1answer
221 views

Conformal Extension from a closed set to open

Let $Q = \{(x,y): x,y\geq 0\} $ be the 1st quadrant of $\mathbb R^2$, and $f$ is a function defined on it such that all the partial derivative(any order) of $f$ exists and continuous. By Whitney ...
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 ...