On the geometry of the symmetrized bidisc

Abstract

We study the action of the automorphism group of the 2 complex dimensional manifold symmetrized bidisc G on itself. The automorphism group is 3 real dimensional. It foliates G into leaves all of which are 3 real dimensional hypersurfaces except one, viz., the royal variety. This leads us to investigate Isaev's classification of all Kobayashi-hyperbolic 2 complex dimensional manifolds for which the group of holomorphic automorphisms has real dimension 3 studied by Isaev. Indeed, we produce a biholomorphism between the symmetrized bidisc and the domainIsaev calls it D1. The road to the biholomorphism is paved with various geometric insights about G.Several consequences of the biholomorphism follow including two new characterizations of the symmetrized bidisc and several new characterizations of D1. Among the results on D1, of particular interest is the fact that D1 is a "symmetrization". When we symmetrize (appropriately defined in the context in the last section) either Ω1 or D(2) 1 (Isaev's notation), we get D1. These two domains Ω1 and D(2) 1 are in Isaev's list and he mentioned that these are biholomorphic to D × D. We produce explicit biholomorphisms between these domains and D × D.