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

TL;DR

This paper characterizes the bidisc by examining a special subgroup of its automorphism group, showing that certain hyperbolic manifolds with similar automorphism properties are biholomorphic to the bidisc.

Contribution

It establishes a new characterization of the bidisc based on the structure of a specific subgroup of its automorphism group, linking geometric and group-theoretic properties.

Findings

The automorphism group of the bidisc contains a subgroup isomorphic to the disc's automorphism group.

Such subgroups induce a partition of the bidisc into a complex curve and pseudo-convex hypersurfaces.

Manifolds with similar automorphism subgroup structures are biholomorphic to the bidisc.

Abstract

We make a connection between the structure of the bidisc and a distinguished subgroup of its automorphism group. The automorphism group of the bidisc, as we know, is of dimension six and acts transitively. We observe that it contains a subgroup that is isomorphic to the automorphism group of the open unit disc and this subgroup partitions the bidisc into a complex curve and a family of strongly pseudo-convex hypersurfaces that are non-spherical as CR-manifolds. Our work reverses this process and shows that any $2$-dimensional Kobayashi-hyperbolic manifold whose automorphism group (which is known, from the general theory, to be a Lie group) has a $3$-dimensional subgroup that is non-solvable (as a Lie group) and that acts on the manifold to produce a collection of orbits possessing essentially the characteristics of the concretely known collection of orbits mentioned above, is biholomorphic to the bidisc. The distinguished subgroup is interesting in its own right. It turns out that if we consider any subdomain of the bidisc that is a union of a proper sub-collection of the collection of orbits mentioned above, then the automorphism group of this subdomain can be expressed very simply in terms of this distinguished subgroup.