New Phenomena on Bounded Nonhomogeneous Domains

In this paper we get the invariant functions and the invariant harmonic func tions under Aut (D) for certain Reinhardt domains D. By using the invariant functions, we get much more invariant Kähler metrics. And the Ricci curvatures, scalar curvatures and holomorpic sectional cur vatures are also obtained, which are very different for the bounded homogeneous domains.