Cohomogeneity One Metrics On Two Pseudoconvex Domains

In this thesis,we mainly study the Bergman metric and cohomogeneity one metric on two pseudoconvex domains.The full text is divided into two chapters.In the first chapter,we introduce the historical background,the related preliminary knowledge and the main results.In the second chapter,the two domains we study are the symmetrized bidisc G2 and an unbounded pseudoconvex domain Ω.The unbounded pseudoconvex domain Ω is a special case of the twisted Fock-Bargmann-Hartogs domain,whose automorphism group is noncompact.First of all,we introduce the analytic and geometric properties of these two domains.Secondly,using the Cartan function,we construct an auxiliary function,which is invariant under the holomorphic automorphism group,and obtain the Kahler metric with the cohomogeneity one potential function of G2.Especially,we obtain the Bergman metric on G2.Thirdly,we use two methods to calculate the Bergman metric of Ω,and then we calculate its holomorphic sectional curvature and the holomorphic bisectional curvature.Finally,we transform the Monge-Ampere equation of the KahlerEinstein metric of Ω into an ordinary differential equation,and then we get the implicit expression of cohomogeneity one metric on some subdomains of this domain.