Holomorphic maps: Geometric aspects

Abstract

We study holomorphic, in particular biholomorphic, maps between domains in c and, in one case, in cⁿ, 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”.