In this paper,we consider some circular domains.And we give an extension theorem for some normalized biholomorphic convex mapping on some circular domains.Especially,we discover the normalized biholomorphic convex mapping on some circular domains have the form f(z) =(f1(z1),...,fn(zn)),where fj:D → C are normalized biholomorphic convex mapping.
In 1965, Lu Yu-Qian discovered that the Poisson kernel of the homogenous domain S m,p,q={Z∈Cm×m, Z1∈Cm×p,Z2 ∈Cq×m|2i1( Z-Z+)-Z1Z1′-Z2′Z2>0} does not satisfy the Laplace-Beltrami equation associated with the Bergman metric when S m,p,q is not symmetric. However the map T0:Z→Z, Z1→Z1 , Z2→Z2 transforms S m,p,q into a domain S I (m, m + p + q) which can be mapped by the Cayley transformation into the classical domains R I (m, m + p + q). The pull back of the Bergman metric of R I (m, m + p + q) to S m,p,q is a Riemann metric ds 2 which is not a Khler metric and even not a Hermitian metric in general. It is proved that the Laplace-Beltrami operator associated with the metric ds 2 when it acts on the Poisson kernel of S m,p,q equals 0. Consequently, the Cauchy formula of S m,p,q can be obtained from the Poisson formula.
LU Qi-Keng Institute of Mathematics, Academy of Mathematics and System Science, Chinese Academy of Sciences, Beijing 100190, China
In this paper, we consider the following Reinhardt domains. Let M = （M1, M2,..., Mn） ： [0,1] → [0,1]^n be a C2-function and Mj（0） = 0, Mj（1） = 1, Mj″ 〉 0, C1jr^pj-1 〈 Mj′（r） 〈 C2jr^pj-1, r∈ （0, 1）, pj 〉 2, 1 ≤ j ≤ n, 0 〈 C1j 〈 C2j be constants. Define DM={z=（z1,z2,…,Zn）^T∈C^n：n∑j=1 Mj（｜zj｜）〈1}Then DM C^n is a convex Reinhardt domain. We give an extension theorem for a normalized biholomorphic convex mapping f ： DM -→ C^n.
Let KD(z,z)be the Bergman kernel of a bounded domain D in Cnand SectD(z,ξ)and RicciD(z,ξ)be the holomorphic sectional curvature and Ricci curvature of the Bergman metric ds2=TDαβ(z,z)dzαdzβrespectively at the point z∈D with tangent vectorξ.It is proved by constructing suitable minimal functions that SectD(z,ξ)2-2 n+2n+1 KD1(z,z)TD1αβ(z,z)ξαξβKD2(z,z)TD2λμ(z,z)ξλξμ2and RicciD(z,ξ)n+1-(n+2)KD1(z,z)TD1(z,z)ξαξβKD2(z,z)TD2λμ(z,z)ξλξμ,where z∈D1DD2,D1is a ball contained in D and D2is a ball containing D.
Given a complete ortho-normal system = (0, 1, 2, . . .) of L2H(D), the space of holomorphic and absolutely square integrable functions in the bounded domain D of Cn, we construct a holomorphic imbedding ι : D →■(n, ∞), the complex infinite dimensional Grassmann manifold of rank n. It is known that in ■(n, ∞) there are n closed (μ, μ)-forms τμ (μ = 1, . . . , n) which are invariant under the holomorphic isometric automorphism of ■(n, ∞) and generate algebraically all the harmonic differential forms of ■(n, ∞). So we obtain in D a set of (μ, μ)-forms ι*τμ (μ = 1, . . . , n), which are independent of the system chosen and are invariant under the bi-holomorphic transformations of D. Especially the differential metric ds21 associated to the Khler form ι*τ1 is a Khler metric which differs from the Bergman metric ds2 of D in general, but in case that the Bergman metric is an Einstein metric, ds12 differs from ds2 only by a positive constant factor.
LU QiKeng Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
