Bergman Kernel Function On Hua Domains And Super-cartan Domain Of The Comparison Theorem

In this paper, we compute the Bergman kernel function with explicit formula of the third type of Hua domain HL~j(p1,. . ,p,.,q, N1,.~ ,N~) (orHE,), and prove the comparison theorem for Bergman metric and Kobayashi metric on the third type of Super-Cartan domain Y:(N,q;K)(or Y,). In 1950? L.K.Hua obtained Bergman kernel functions with explicit formulas for four types of irreducible symmetric classical domains by the holomorphic transitive groups(called Hua method). So the basic theory of harmonic analysis of several complex variables on the four types of Cartan domains is built. This is a classical work in the several com- plex variables. For non-symmetric homogeneous domains, we can also get the explicit formulas of their Bergman kernel functions by Hua method. We know the complete or- thonormal system of the Egg domain is made up of monomials, 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 do not use the Hua method or the method of summing a series to get Bergman kernel function. In this paper, we give a tiew method to compute the Bergman kernel function. It con- sists of two steps: first, we give the group of holoniorphic automorphism, such that the 6 element f(w, z) of the group maps (w, z) into (w, 0). Thus, the Bergman kernel function is K((w,z);(w,z)) = Jdet(Jf)j2K((w~,0);(w*,0)), where (.Jj) is the Jacobi matrix off, det(Jf) is the determinant of (Jr). It can be computed easily. It followed that the prob- lem is that we only need compute K((w*, 0); (wy, 0)). Second, we define Semi-Reinhardt domain , and compute the complete orthonormal system of Semi-Reinhardt domain. But Flua domain is Semi-Reinhardt domain, and by the complete orthonormal system, we know K((w~,O);(w,O)) is a multi-infinite series about (wt, .. . ,w~), 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 third type of Hua domain is: r j= 1 where llw()112 lw,i 2 + ... + 1w3N, 12, N1, . . . , N~, q are positive integers; p1, ... 1p~ are positive, Pju(q) is the third type of symmetric classical domain. By computation, we know the Bergman kernel function of the third type of Hua domain is: Cdet(l + ~~y~~1=1 ~YL~q ~ pi ? k1+1 ~ k1 k,- ~ +1 V~ I Where C = = IIW(j)~ = llw(,)ll det(I + ZZ)11~2,j = 1,2,... ,r. this is (2.25)?in chapter two, and Bergman kernel function of flEE in the most average cases. When the sum of the infinite series is obtained, w...