Several Problems On The Generalized Hua Domains

In this thesis, we consider the following three aspects: first, we compute the Bergman kernel functions with explicit formulas on generalized Hna domains ; second, we obtain the explicit formulas for extremal maps and extremal values between the ball and the super-Cartan domain of the first type; finally, we give sufficient conditions and necessary conditions that holomorphic functions become Bloch functions on super-Cartan domains.We have the following results.Part I The computations of the Bergman kernel functionsThe Bergman kernel function plays an important role in several complex variables. S. Bergman introduced the concept of Bergman kernel function in 1921 when he studied the orthogonal expansion on D in C and he generalized it to the case in several complex variables in 1933. It is well known that there exists an unique Bergman kernel function for each bounded domain in Cn. For which domains can the Bergman kernel function be computed by explicit formulas? This is an important problem. Explicit formula of the Bergman kernel function can help us to solve many important conjectures. We illustrate this point by two case. Mostow and Siu have given a counterexample to the important conjecture that the universal covering of a compact K(?)hler manifold of negative sectional curvature should be biholomorphic to the ball. In their counterexample the explicit calculation of Bergman kernel function and metric of the egg domain {z C2 : |z1|2+ |z2|14 < 1} plays an essential role [MoS]. Another example is about Lu Qikeng conjecture: in order to give a counterexample to the Lu Qikeng conjecture, an explicit formula for the Bergman kernel function is used in ref.[Bo]. Therefore, computation of the Bergman kernel function by explicit formula is an important research direction in several complex variables. Up to now, there are still many mathematicians working in this direction.But before Yin Weiping constructed Hua domains, only there are two types of domains on which the Bergman kernel functions can be obtained explicitly: the bounded homogeneous domains and the complex ellipsoid domains in some cases.Hua Lookeng obtained Bergman kernel functions with explicit formulas on four types of irreducible symmetric classical domains by the holomorphic transitive groups (called Hua method) [Hu1][Hu2]. For some non-symmetric homogeneous domains, we can also get the explicit formulas of their Bergman kernel functions by Hua method [Xu4][Gi].We know the complete orthonormal system of the bounded Reinhardt domain made up of monomials, and complex ellipsoid domain is the bounded Reinhardt domain, so the explicit formulas of the Bergman kernel functions are obtained by summing an infinite series in some cases (called method of summing series).In general, it is difficult to get, the domain whose Bergman kernel function can be gotten explicitly. So some mathematicians think the domain with explicit Bergman kernel function is worth researching and is a good domain.Yin Weiping constructed a new type of domain with explicit Bergman kernel function in 1998 and generalized continuously them from that time. He constructed the following four types of domains in 2001, called generalized Hua domains:where wj = (wj1,...,WjNj), j = 1, ... ,r. 1(m,n), II(p), III(q), (n) denote respectively the Cartan domains of the first type, second type, third type and fourth type in the sense of L. K. Hua, ZT denotes the conjugate and transposed of Z and det denotes the determinant of a square matrix. N1 , ... , Nr; p1-1 , ... ,pr-1-1 are positive integers and pr,k are positive real numbers. When k = 1 , they are the Hua domains.The generalized Hua domains are neither homogeneous domains nor Reinhardt domains. So we can not, use the Hua method or the method of summing a series to get Bergman kernel functions in explicit forms. We give a new method to compute the Bergman kernel functions on generalized Hua domains. It consists of two steps: first, we give the group of holomorphh: automorphism on every typ...