Hua Domain Of The Same Measure

In this thesis, we mainly discuss the invariant metrics on the third Hua domains. There are three topics,the Einstein-Kahler metric on the third Cartan-Hartogs domain, the Bergman kernel function on the third Hua Construction and the equivalence among the classical metrics on the third Cartan-Hartogs domain.In the first chapter, we discuss the Einstein-Kahler metric on the third Cartan-Hartogs domain. We get the complete Kahler-Einstein metric on the third Cartan-Hartogs domain. The third Cartan-Hartogs domain is well defined as follows:Y_III(n,q;K) = {w∈Cⁿ,Z∈ R_III(q) : ||w||² < det(I - ZZ^T)^1/K,K > 0} := Y_III,where R_III(q) is the third type of symmetric classical domain in the sense of L.K. Hua[1], and det is the abbreviation of determinant, q > 1 is a positive integer. w is a vector with n entries and ||w||² = |w₁|² + |w₂|² + … + |w_n|². The Bergman kernel function on the third Cartan Hartogs domain Ym(n, q; K) has been given in explicit formula^[2] and we can know that Y_III is Bergman exhaustion. Therefore, Y_III is bounded pseudo convex domain[3,4]. Cheng-Yau[5] and Mok-Yau[6] proved that every arbitrary bounded pseudoconvex domainΩ in Cⁿ admits a complete Kahler-Einstein metric.If this Einstein-Kahler metric is given byand g is the unique solution to the boundary problem of the complex Monge-Ampere equation:then g is called generating function of E_d(z). Obviously, if one obtains g in explicit formula, then the Einstein-Kahler metric is also explicit. But it is very difficult to get the explicit solution of the Monge-Ampere equation, because the equation is completely nonlinear. We adopt a special method by using the holomorphic automorphism groupof the third Cartan-Hartogs domain and the invariant founction X under holomorphic automorphism to reduce the Mong-Ampere into the ordinary differential equation, then we get the implicit solution of the generating function of the Einstein-Kahler metric on the third Cartan-Hartogs domain. When the parameter K =(q/2)+(1/(q-1)) we get the complete Kahler-Einstein metric with explicit form. And when q > 2 the third Cartan Hartogs domain Y_III(n, q, (q/2) + 1/(q-1)) is a non-homogeneous domain. We have computed the holomorphic sectional curvature of the complete Einstein-Kahler metric and obtained a sharp estimation of it. Due to the result of M. Heins[7], we have proved that comparison theorem between the Kahler-Einstein metric and the Kobayashi matric. Finally, the equivalence between complete Einstein-Kahler metric and Bergman metric on Y_III(n, q; K) is described as an application.In the second chapter, we discuss the equivalence among the classical metrics on the third Cartan-Hartogs domain. The most important result in this chapter is .that we prove an old conjecture of Yau about the equivalence of the Kahler-Einstein metric and the Bergman metric on an irreducible domain covered with a compact Kahler manifold. We use a very different method to get the equivalence of the Kahler-Einstein metric and the Bergman metric. Firstly, we introduce a new metric and then we prove that this new meteic is equivalent to the Bergman metric on the third Cartan-Hartogs domain. Then we know this new meteic is complete. Secondly we get the negative lower and upper estimations of the Ricci curvature and holomorphic sectional curvature of this new meteic. With the Schwarz-Yau lemma[8], we can prove the equivalence of the new metric and the Kahler-Einstein metric on third Cartan-Hartogs domain. Finally , by using the new metric as a bridge we can prove that the Bergman metric is equivalent to the Einstein-Kahler metric on the third Cartan-Hartogs domain. Especially, we can get the equivalence among the four classical metrics when the third Cartan-Hartogs domain is convex.In the third chapter, we discuss the explicit formula of the Bergman kernel functionon the third Hua Construction. The third Hua Construction is defined as follows:where R_III(q) denotes the Cartan domain of the third type. In this chapter we obtain the explicit Bergman kernel function for HC_III when 1/(p1), …,1/(pn-1) are positive integers and — is any positive real number. According to the definition and the transformation formula of the Bergman kernel function, We take use of the holomorphic automorphism group Aut(HC_III) and the complete orthonormal system on HC_III to compute a multi-infinite series instead of computing the Bergman kernel function. And with some skills we get the sum of the series when 1/p1,…,1/(pn-1) are positive integers and -1/pn is any positive real number. So we have the explicit Bergman kernel function on third Hua Construction.