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By L.; Nakai, M. Sario

The goal of the current monograph is to systematically boost a class idea of Riemann surfaces. a few first steps may also be taken towards a type of Riemannian areas. 4 levels will be exotic within the chronological heritage: the sort challenge; normal type; compactifications; and extension to better dimensions. the kind challenge developed within the following a little bit overlapping steps: the Riemann mapping theorem, the classical style challenge, and the lifestyles of Green's features. The Riemann mapping theorem laid the root to type idea: there are just conformal equivalence sessions of (noncompact) easily hooked up areas. Over part a century of efforts by way of top mathematicians went into giving a rigorous evidence of the concept: RIEMANN, WEIERSTRASS, SCHWARZ, NEUMANN, POINCARE, HILBERT, WEYL, COURANT, OSGOOD, KOEBE, CARATHEODORY, MONTEL. The classical variety challenge was once to figure out even if a given easily attached masking floor of the airplane is conformally resembling the aircraft or the disko the matter was once within the heart of curiosity within the thirties and early forties, with AHLFORS, KAKUTANI, KOBAYASHI, P. MYRBERG, NEVANLINNA, SPEISER, TEICHMÜLLER and others acquiring incisive particular effects. the most challenge of discovering precious and enough stipulations is still, notwithstanding, unsolved.

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