Oberseminar Funktionentheorie, The dark side of the unit disk II: The geometry of Omega

In this talk, we investigate the geometric features of Ω:={(z,w)∈ℂ̂²:zw≠1} in its various incarnations. As an appetizer, we convince ourselves that any holomorphic mapping with an analytic continuation beyond Ω⊆ℂ̂² is constant. Afterwards, we work towards characterizing the subgroup of automorphisms of Ω preserving its Laplacian. Quite surprisingly, it turns out to be a double covering of the Möbius group and thus in particular a finite dimensional Lie group. The key ingredient of the proof is the classical Schwarzian derivative, for which we will discuss a holomorphic difference quotient. Moreover, we introduce the holomorphic analogue of the hyperbolic and spherical metrics, endowing Ω with the structure of a holomorphic Kähler manifold. Matching with this, we show that one may regard Ω as a homogeneous space over its automorphism group. Finally, the domains Ω_± make their first appearance.