A characterization of the bidisc by a subgroup of its automorphism group
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 pseudoconvex hypersurfaces that are nonspherical as CRmanifolds. Our work reverses this process and shows that any $2$dimensional Kobayashihyperbolic manifold whose automorphism group (which is known, from the general theory, to be a Lie group) has a $3$dimensional subgroup that is nonsolvable (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 subcollection 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.
 Publication:

arXiv eprints
 Pub Date:
 April 2021
 arXiv:
 arXiv:2104.05001
 Bibcode:
 2021arXiv210405001B
 Keywords:

 Mathematics  Complex Variables;
 Primary 32M05;
 Secondary 32A07
 EPrint:
 10 pages, To appear in J. Math. Anal. Appl