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If one deals with a simply-connected 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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I talked to my advisor about this and he doesn't think there is much. The fact that having the bergman kernels allows you to construct the biholomorphism in 1 complex variable is really kind of a fluke: The "change of coordinates" formula for the bergman kernel in higher dimensions involves the determinant of the complex Jacobian. It is pretty rare that you would somehow know that two domains were biholomorphic without having an explicit map in higher dimensions: we don't have anything like a Riemann mapping theorem. –  Steven Gubkin Sep 18 '12 at 0:10

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