## Abstract

We study holomorphic, in particular biholomorphic, maps between domains in c and, in one case, in c^{ n }*, n >* 1*.* These maps are for *n* = 1 *conformal* (angle and orientation preserving); so we shall use the terms *biholomorphic* and *conformal* interchangeably in this case. For *n >* 1 we consistently use *biholomorphic*. *Automorphisms* of domains, i.e. biholomorphic self-maps, are determined for disks resp. half-planes, the entire plane, and the sphere: they form groups consisting of Möbius transformations (VII.1). The proof of this fact relies on an important growth property of bounded holomorphic functions: the Schwarz lemma 1.3. Because the automorphism group of the unit disk (or upper half plane) acts transitively, it gives rise – according to F. Klein's Erlangen programme – to a geometry, which turns out to be the *hyperbolic (non-euclidean)* geometry (VII.2 and 3). The unit disk is conformally equivalent to almost all simply connected plane domains: Riemann's mapping theorem, proved in VII.4. For *n >* 1 even the immediate generalisations of the disk – the *polydisk* and the *unit ball* – are not biholomorphically equivalent (VII.4). Riemann's mapping theorem can be generalised to the general uniformization theorem (VII.4): a special but exceedingly useful case of this is the *modular map λ* which we introduce in VII.7. Its construction uses tools that are also expedient for other purposes: *harmonic functions* (with a solution of the Dirichlet problem for disks) and Schwarz's reflection principle (VII.5 and 6). The existence of *λ* finally yields two important classical results: Montel's and Picard's “big theorems”.

## Keywords

Harmonic Function Holomorphic Function Unit Disk Geometric Aspect Hyperbolic Geometry## Preview

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