The Equivalent Classical Metrics on the Cartan-hartogs Domains

In this paper we study the complete invariant metrics on CartanHartogs domains which are the special types of Hua domains. Firstly, we introduce a class of new complete invariant metrics on these domains, and prove that these metrics are equivalent to the Bergman metric. Secondly, the Ricci curvatures under these new metrics are bounded from above and below by the negative constants. Thirdly, we estimate the holomorphic sectional curvatures of the new metrics, we prove that the holomorphic sectional curvatures are bounded from above and below by the negative constants. Finally, by using these new metrics and Yau’s Schwarz lemma we prove that the Bergman metric is equivalent to the Einstein-Kähler metric. That means the Yau’s conjecture is true on Cartan-Hartogs domain. The concept of Hua domain was introduced by Weiping Yin in 1998. Since then, many good results have been obtained. The Bergman kernel functions are given in explicit forms[1-17]. The comparison theorems for Bergman metric and Kobayashi metric are proved on Cartan-Hartogs domains[18-21]. The explicit form of the Einstein-Kähler metric is got on non-symmetric domain which is the first time in the world[22-26], etc. In this paper we will study the equivalence between the classical metrics. There are many deep results on this subject. Let ωB(D), ωC(D), ωK(D), ωEK(D) be the Bergman metric, Carathéodory metric, Kobayashi metric and Einstein-Kähler metric on bounded domain D in C respectively. Then we have ωC(D) 6 2ωB(D)[27,28], ωC(D) 6 ωK(D)[29], ωC(D) = ωK(D) if D is the convex domain[30], ωB(D) = ωEK(D) if D is the bounded homogeneous domain in C[31,P.300]. For the ωB(D) and ωK(D), no relationship is known. People had hoped that the inequality ωB(D) 6 CωK(D) for some universal constant C would hold, but in 1980 Diederich and Foraess [32] showed that there exist pseudoconvex domain in C where the quotient ωB(D)/ωK(D) is unbounded. If the inequality ωB(D) 6 CωK(D) holds then we say that the comparison theorem for Bergman Date: 18th September 2005. 2000 Mathematics Subject Classification. 32H15, 32F07, 32F15.