On the geometry of the symmetrized bidisc

Abstract

We study the action of the automorphism group of the $2$ complex dimensional manifold symmetrized bidisc $\mathbb{G}$ on itself. The automorphism group is 3 real dimensional. It foliates $\mathbb{G}$ 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 domain \[\{(z_1,z_2)\in \mathbb{C} ^2 : 1+|z_1|^2-|z_2|^2>|1+ z_1 ^2 -z_2 ^2|, Im(z_1 (1+\overline{z_2}))>0\}\] in Isaev's list. Isaev calls it ${\mathcal{D}}_1$. The road to the biholomorphism is paved with various geometric insights about $\mathbb{G}$. Several consequences of the biholomorphism follow including two new characterizations of the symmetrized bidisc and several new characterizations of ${\mathcal{D}}_1$. Among the results on ${\mathcal{D}}_1$, of particular interest is the fact that ${\mathcal{D}}_1$ is a "symmetrization". When we symmetrize (appropriately defined in the context in the last section) either ${\Omega }_1$ or ${\mathcal{D}}_1^{(2)}$ (Isaev's notation), we get ${\mathcal{D}}_1$. These two domains ${\Omega }_1$ and ${\mathcal{D}}_1^{(2)}$ are in Isaev's list and he mentioned that these are biholomorphic to $\mathbb{D}\times \mathbb{D}$. We produce explicit biholomorphisms between these domains and $\mathbb{D}\times \mathbb{D}$.

22 pages, Accepted in Indiana University Mathematics Journal

References in corpus (1)

• A Geometric Characterization of the Symmetrized Bidisc

Cited by in corpus (1)

• A characterization of the bidisc by a subgroup of its automorphism group