Conformal structure of regions of the complex plane and the ring of holomorphic functions

How is the conformal structure of regions of the complex plane determined by the integral domain of holomorphic functions defined on those regions?

Thanks

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I understand all the words, but I still don't understand the question: what you do mean by "determining" the conformal structure? Also some context would help. – Pete L. Clark May 20 '11 at 3:46
Ah, here's a guess: are you asking whether any Riemann surface $X$ with field of meromorphic functions isomorphic to the field of meromorphic functions on $\mathbb{C}$ is actually isomorphic to $\mathbb{C}$? (That would be a good question...) – Pete L. Clark May 20 '11 at 3:52
I mean determined in the sense that given two regions of the complex plane such that there is an isomorphism between the fields of meromorphic functions defined on each region, then these regions will be conformally equivalent. – user7485 May 20 '11 at 3:59
.: okay, that's a perfectly good thing to mean. It doesn't seem to be what you said though, so you should probably edit your question for clarity. (And if this is what you mean, why is your question tagged riemann-surface?) – Pete L. Clark May 20 '11 at 4:42
Thanks Pete, I've edited the question- it turns out that it's the integral domain of holomorphic functions that determines the conformal structure of the regions (though in general, plane regions), and the field of meromorphic functions determines the conformal structure of Riemann surfaces. – user7485 May 20 '11 at 9:07