Biholomorphic Mappings Between Isodimensional Quasiconvex Regions

The study of holomorphic equivalence is one of the most important research topics in several complex variables,in which the rigidity of biholomorphic mappings between bounded pseudoconvex domain is a classical research branches.In this paper,we explore the rigidity of holomorphic mappings between equidimensional pseudoconvex domain.More precisely,we mainly focus our attention on the Hartogs domains.Firstly,we discuss the biholomorphic equivalence problem of a special Hartogs domain called Hua domain(named after Loo-Keng Hua),which is a generalization of Cartan-Hartogs domain.Hua domain can be regarded as a Hartogs domain where the base is a bounded symmetric domain and the fiber is a complex ellipsoid.In general,a Hua domain is a nonhomogeneous bounded pseudoconvex domain without smooth boundary.We introduce the function Y?(z,w)(see Equation 2-(1))on Hua domain.By using this function,we give a sufficient and necessary condition for the two Hua domains to be biholomorphic euquivalent.In particular,we obtain the equivalent characterization for the holomorphic automorphism of the Hua domains.Secondly,we study the generalized Fock-Bargmann-Hartogs domains with Cn as its base.In general,each generalized Fock-Bargmann-Hartogs domain is an unbounded non-hyperbolic pseudoconvex domain without smooth boundary.With the help of the function L(z,w)(see Equation 3-(1)),we find that the rigidity of the holomorphic automorphism of the generalized Fock-Bargmann-Hartog domains can be established.