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Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.
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How to produce a biholomorphism
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 …
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Bounded mappings on unbounded domains
Ahlfors and Beurling (Ahlfors, Lars; Beurling, Arne Conformal invariants and function-theoretic null-sets. Acta Math. 83, (1950). 101–129.) provided an explicit example of a domain in $\mathbb C$ whic …