The Bergman Kernel Function On The Hua Domain, The Comparison Theorem And Einstein-K(?)hler Metric On Super-Cartan Domain Of The First Type

In this thesis, we compute the Bergman kernel function with explicit formula on Hua domain of the first type HE;(N1,.... ,Nr.m.n; p],....pr), and prove the comparison theorem for the Bergman metric and Kobayashi metric on Sviper-Cartan domain of the first type Yf(N, m. n; K), and get the Einstein-Kahler Metric on Super-Cartan Domain of the first type F/(l,m, n; K).We have the following results.Part IThe Bergman kernel function plays an important role in several complex variables. The concept of Bergman kernel function was introduced by S. Bergman in 1921. It is well known that there exists an unique Bergman kernel function for each bounded domain in C? 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 important conjectures. We illustrate this point by two case. Mostow and Sin have given a counterexample to the important conjecture that the universal covering of a compact Kahler 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 G C2 : Z]\2 + z?}14 < 1} plays an essential role[36]. Another example is 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. [5]. Therefore, computation of the Bergman kernel function by explicit formula is an important researchdirection in several complex variables. Up to now, there are still many mathematicians working in that direction[3][10][ll][15][24][43][60].Bnt we can get the Bergman kernel function in explicit formulas for a few types of domains only, for example: the bounded homogeneous domains and the complex ellipsoid domains in some case.Hua Lookeng obtained Bergman kernel functions with explicit formula of four types of irreducible symmetric classical domains by the holomorphic transitive groups (called Hua method). For some non-symmetric homogeneous domains, we can also get the explicit formulas of their Bergman kernel functions by Hua method[44][45][46j.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. By now, we can compute the explicit formulas of the Bergman kernel functions on the upper two types of domains.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 constructs a new type of domain with explicit Bergman kernel function. It is called Hua domain. The domain is neither homogeneous domain nor Reinhardt domain. So we can not use the Hua method or the method of summing a series to get Bergman kernel function. In first part, we give a new method to compute the Bergman kernel function. It consists of two steps: first, we give the group of holomorphic automorphism, such that the element F(w,z) of the group maps (w,z) into (w*.0). Thus, the Bergman kernel function is K((w,z):(w.z)} = |det(JF)|2K((w*,0); (w*.0))< where (JF) is the Jacob! matrix of F, det(Jf] is the determinant of (./F). It can be computed easily. It followed that the problem is that we only need compute K((w*,Q); (w*.0)). Second, we define Serni-Reinhardt domain, and compute the complete orthonormal system of Semi-Reinhardt domain. Because Hua domain is Semi-Reinhardt domain, and by the complete orthonormal system, we know K( ) is a multi-infinite series about . then we can get the Bergman kernel function by summing the infinite series. After the sum is obtained, we think we get the explicit formula.The definition of the first type of Hua domain is:By computation, we know the Bergman kernel function of...