Interesting... How do you go about rigorously proving statements like, "If there is more than one way to glue a disk, you must have a Riemann sphere", and "If there is always exactly one way, you can take the picture and put it step-by-step on a complex plane." ? I suppose, if these two are done, then the rest is the Riemann mapping theorem .. maybe that is what Gerald Edgar above had in mind.

A good reference?

Please skip the proof that these three are not conformally equivalent! :)

How does it become a triviality, if Riemann mapping theorem is assumed?

Yes of course .. :) You know what I had mainly in mind; it is the other part, that every simply connected Riemann surface is conformally equivalent to one of the above three.

The two answers do look somewhat different, though I can't put my finger exactly on what it is.

Oh so it must use formal groups. I tried to look into the book of Hazewinkel; but it is too big for me. The book of Serre on Lie Algebras and Lie Groups seem to explain this notion better. What is an easy introduction to formal groups? What is the difference between a formal group and a formal group law? Must such questions on abelian varities use formal groups for solution? And to what extend can you recover an abelian variety/algebraic group/lie group if you know its formal group? Perhaps is it better to ask this question separately?

I understand. How about, if the student is facing funding problems? This is the economic problem you are talking about. And I must motivate him to look for new positions, among other stuff. I suppose the book is available in public domain by now? Is it available for free somewhere?

Yeah but so many nice answers are coming. If you remove it, please allow it a couple of days to stay at least. Anyway I asked. Let me at least hear some response. With this moderator meta comment, people completely stopped contributing any answers. It is a real pity. The answers all given here are useful. What is the need for such a haste to ban this question?