If one deals with a simplyconnected domain in the complex plane which is not the whole plane then it is easy to construct the biholomorphism mapping it to the unit disc. This can be done by means of the Bergman kernel and the construction is as "explicit" as is the kernel. My question is about dimension higher than one. Given two domains for which one knows in advance that they are biholomorphic, are there any methods (I don't know maybe sheaf theoretic or using $\bar\partial$ theory) or procedures to obtain the biholomorphism between them? Most of the literature deals with the problem of distinguishing when two domains are not biholomorphic so it is not helpful.
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