Automorphism groups of domains that depend on fewer than the maximal   number of parameters
  We study domains in complex $n$-space with automorphism group that does not depend on the full $n$ dimensions of the ambient space. A sufficient geometric condition is obtained to guarantee that a domain has such a "thin" automorphism group. Examples are provided to illustrate the ideas. 
Jisoo Byun and Steven G. Krantz