Alladi Ramakrishnan Hall

Classifying domains in $C^n$ by their automorphism group: a short survey and a case study of domains with abelian automorphism group

G. P. Balakumar

IMSc

The problem of classification of domains in $C^n$ upto
biholomorphic equivalence is of fundamental importance in complex
analysis. From the Riemann Mapping Theorem (RMT), one may derive the fact
that every smoothly bounded homogeneous planar domain is equivalent to the
unit disc. This fact continues to hold in higher dimensions, even though
the RMT fails in every dimension n>1. This and several other positive
classification results are possible; collectively leading to the belief
that the classification problem, when formulated appropriately, is
tractable in multi-dimnesional complex analysis as well.

In this talk, I shall review and briefly discuss such results. With this
background built, I shall report a few recent results and conclude with my
own contributions concerning model domains with abelian automorphism group
-- a part of my thesis work.