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 a mapping is unique.

Now I'm interested in the related case of the infinite dimensional Lie group $Conf(\mathbb H^2)$ that maps the upper half plane to itself. The Lie group is locally isomorphic to $Diff(S^1)$ (I think... some details may vary), so it should be clear that there are subsets $B' \subset \mathbb H^2$ that cannot be mapped from $B$ by a $Conf(\mathbb H^2)$ transformation.

Specifically, I'm interested in some kind of intuition about the possible forms of allowed mappings as above, i.e. some "number" of possible images of such mappings. Clearly the answer will involve $Diff(S^1)$ in some important way...

In short:

-$Diff(\mathbb C)$ maps $B$ to any $B'$ in infinite number of possible ways

-$Conf(\mathbb C)$ maps $B$ to any $B'$ in a unique way

-$Conf(\mathbb H^2)$ maps $B$ to *some* subsets $B'$... but what kind of $B'$s??

I actually have in mind a way to "number" such maps, but there are probably much better mathematical expositions/proofs lying around somewhere...

**EDIT:** OK I just *knew* I should've stuck with infinite dimensional Lie algebras instead of Lie groups (as per Robert Bryant's comment below)... so here are some corrections (I'll leave the above stuff intact for the sake of my own public humiliation):

1) Yes indeed I meant a proper subset which is connected and simply connected

2) OK so the Riemann Mapping is unique up to $PSL(2;\mathbb R)$... didn't realize that!

3) So instead of "$Conf(\mathbb H^2)$" let's think about holomorphic vector fields on $\mathbb H^2$, e.g.

$V = \xi(z) \partial_z + \xi(\bar z) \partial_{\bar z}$.

As $Im(z) \to 0$, these tend to $\xi(x) \partial_x \in Vect(S^1)$ (the boundary of $\mathbb H^2$ is $S^1$). I think it's safe to say that there are flows at least for some $\xi$ such that the resulting mapping is a conformal transformation? That's what I meant by $Conf(\mathbb H^2)$, which was probably wrong in many ways...

So the question then applies to *flows* of $V$... I hope it's clearer now!

propersubset of $\mathbb{C}$, you left out the 'connected' hypothesis (unless you include that in your definition of 'simply connected'), and even then, the mapping is never unique; it is only unique up to composition with an element of $Conf(B)\simeq \mathrm{PSL}(2,\mathbb{R})$. Similarly, you seem not to realize that $Conf(\mathbb{H}^2)$ isnotinfinite dimensional at all, but is, in fact, of dimension $3$. Try again? $\endgroup$3more comments