Geometric Analysis On The Hartogs Domains In Several Complex Variables

Hartogs domain is the classical subject of complex analysis in several variables.It is mainly divided into two parts:the base domain and the fiber at each point in the base domain.In fact,Hartogs domain may inherit some geometric properties of the base domain:but generally speaking,it is quite different from the base domain.Therefore,the research on Hartogs domain can be regarded as the extension and deepening of the research on the base domain.Hence,compared with the base domain,Hartogs domain has more abundant research background and deeper research results.In this thesis,we mainly focus on two kinds of special Hartogs domains:Fock-Bargmann-Hartogs domain and the generalized Cartan-Hartogs domain.Actually,we study Kobayashi pseudometric in Fock-Bargmann-Hartogs domain and Berezin quantization in the gen-eralized Cartan-Hartogs domain,respectively.Focus on these two problems,we will divide this thesis into four chapters.In chapter 1,the background,the recent developments:and the main results of this thesis are introduced.In chapter 2,some basic definitions and some basic lemmas are introduced.In chapter 3,we use factorization of bounded holomorphic functions and "station-ary" to build the connection between Fock-Bargmann-Hartogs domain and Siegel half plane,and then we can give the explicit expression of geodesics of Dn,1 in the sense of Kobayashi pseudometric.Thus,by means of the formula of geodesics,after a se-ries of complicated calculations and comparisons,we calculate explicitly the Kobayashi pseudometric on D1,1.Lastly,we establish the Schwarz lemma at the boundary for holo-morphic mappings between the non-equidimensional Fock-Bargmann-Hartogs domains by using the formula for the Kobayashi pseudometric on D1,1.In chapter 4,we introduce a new Kahler potential,?(z,?):=-?jk=1 vj ln NGj(zj,zj)?j-ln(?jk=1 NGj(zj,zj)?j-???2)the generalized Cartan-Hartogs domain(?jk=1Gj)Bd0(?),and then we can obtain a Kahler metric g(?;v)with respect to the Kahler potential?(z.=,?)on the generalized Cartan-Hartogs domain(?jk=1 Gj)Bd0(?).Hence,we will compute the explicit expression of the Rawnsley's ?-function of the generalized Cartan-Hartogs domainn(?jk=1 Gj)Bd0(?)with the Kahler metric g(?;v).Then by further calcu-lating and studying the form of Rawnsley's ?-function,we give necessary and sufficient conditions for ?(?,g(?;v))to become a polynomial in(1-???2).Lastly,we prove that the generalized Cartan-Hartogs domain(?jk=1 Gj)Bd0(?)can admit a Berezin quantization.