How can you tell if a conformal mappimg between regions is unique? I have a conformal mapping from {z : |z|<2, |Arg(z)|< pi/6} to {z : Re(z)>0, Im(z)<0} as

f(z) = -(iz^3 - 8)/(iz^3 + 8) but have no idea how to find out if the conformal mappimg is unique? Any help is much appreciated.

  • $\begingroup$ for uniqueness (of conformal maps between simply connected domains) you need to fix the following pieces of data: $z_0$ in the original domain, $w_0$ in the target domain, $\theta$ an angle; then there is unique conformal $f$ from one domain onto the other with $f(z_0)=w_0, \arg f'(z_0)= \theta$; note that $|f'(z_0)|$ depends on the geometric shapes of the domains and the choice of the image point $w_0$ $\endgroup$ – Conrad Mar 25 at 22:18

Such a conformal mapping will not be unique in general.

For example, suppose $A$ and $B$ are two nonempty, proper, simply connected open subsets of $\Bbb C$, and suppose that $f$ is a conformal mapping from $A$ to $B$.

By the Riemann mapping theorem, both $A$ and $B$ are conformally equivalent to the open unit disk $D$. In particular, there is a conformal mapping $g\colon A\to D$.

The open unit disk also has a whole bunch of conformal automorphisms, given by Möbius transformations, let $m$ be one such that is not the identity map. (A simple example, if we want to be concrete, is $m(z)=-z$.)

It follows that $f$ and $f\circ g^{-1}\circ m\circ g$ are distinct conformal mappings from $A$ to $B$.

(Indeed, for such domains, the set of conformal mappings from $A$ to $B$ actually forms an infinite group that is isomorphic to the group of conformal automorphisms of $D$.)

  • 1
    $\begingroup$ No need for a weird domain, for instance, for the complement to a generic finite subset of ${\mathbb C}$ consisting of at least 4 points, the group of conformal automorphisms is trivial. $\endgroup$ – Moishe Kohan Mar 25 at 22:26

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.