Hua Domain Research

In this thesis, we discuss four topics in all on Hua domains, including the Bergman kernel, the explicit Kahler-Einstein metric, the equivalence of the classical invariant metrics and the zeros of Bergman kernels.Chapter I The complete Kahler-Einstein metricThe existence of the Kahler-Einstein metric on manifolds was proved by Yau in 1970's. Let M be a compact Kahler manifold. Yau[36, 37] proved that there exists a unique Kahler-Einstein metric on each Kahler class when the first Chern class satisfies C₁(M) = 0 or C₁(M) < 0 and this is not true in general for the case C₁(M) > 0. In 1980, Cheng and Yau [38] proved there arc a great deal non-compact manifolds on which the Kahler-Einstein metrics exist. Especially, they showed that any C² bounded pseudoconvcx domain in Cⁿ admits a complete Kahler-Einstein metric with negative Ricci curvature. In 1983, Mok and Yau [39] generalized this result to arbitrary bounded pseudoconvcx domain in Cⁿ. However, they did not give the explicit form of the metric. In a long time, people knew little about the metric except for some homogeneous domains. In 1986, J. S. Bland [42]showed the generating function of the Kahler-Einstein metric with implicit expression on a type of Reinhardt domain {|z|²+ |w|^2p< 1}. In 2001, Yin Weiping and Wang An [31]havc given the complete Kahler-Einstein metric with explicit form on the first type Cartan-Hartogs domain when the fiber dimension is one, i.e. w ∈ C. The definition of the first type Cartan-Hartogs domain is as following.where (Z|-)^t denotes the transpose and conjugate of the matrix Z, (?)(m,n) is the first type symmetric classical domain in the sense of Hua.In this Chapter, I will show the explicit Kahler-Einstein metric on the second type Cartan-Hartogs domain Y_II(N.p.K) when the dimension of the fiber N > 1 and the parameter K = p/2+1/(p+1). In this case, I have computed the holomorphic sectional curvature of the metric and obtained a sharp estimation of it. Due to the result of M. Heins[106], I have proved that comparison theorem between the Kahler-Einstein metric and the Kobayashi matric. Finally. I will show that the Kahler-Einstein metric and the Bergman matric are equivalent when K-P/2+1/(P+1) The definition of Y_II(N.p. K) iswhere (?)_II(p) is the second type symmetric classical domain in the sense of Hua. ||w||²=∑_i=1^N|w_j|².p∈N.Chapter II The equivalence of the classical invariant metricsIt is well known the Bergman metric(B), Caratheodory metric(C), Kobayashi metric(K) and Kahler-Einstein metric(ε) are the four classical biholomorphic invariants in complex analysis. They are great helpful in studying the boundary geometry of the domain and biholomorphic mappings extending smoothly to the boundaries of the relevant domains. Therefore, many mathematicians pay their attention to the equivalence problem of the above four invariant metrics.For instance, S. Kobayashi[15] proved in 1976 that C ≤ K holds for all manifolds. K. T. Hahn[19] showed the relation C ≤2B in 1978. In 1982, L. Lempert proved C = K. on all convex doamins. In 2000, Yin Weiping, Wang An et.al proved B≤C*K, holds both for some Reinhardt domains and Hartogs domains. Recently, Liu Kefeng, Sun Xiaofeng and Yau Shingtong [61] proved that the classical invariant metrics are equivalent on the Teichmuller space and the moduli space of the Riemann surfaces. This solved an old conjecture of Yau about the equivalence of the Kahler-Einstein metric and the Bergman metric.The main result in Chapter II is that we have proved the equivalence of the Kahler-Einstein metric and the Bergman metric also holds on the Cartan-Hartogs domain of the second type for arbitrary K > 0 and the fiber (?) ∈ N. After introducing a new metric which is equivalent to the Bergman metric, we get the negative lower and upper estimations of its Ricci curvature and holomorphic sectional curvature. Then by proving the equivalence of the new metric and the Kahler-Einstein metric in terms of Schwarz-Yau lemma, we obtain the equivalence of the Kahler-Einstein metric and the Bergman metric. As an application, we give a sufficient condition of the four classical invariant metrics equivalence on the Cartan-Hartogs domain of the second type Y_II(r.p,K).Chapter III The Bergman kernel of the Hua constructionIt is an important research field in the several complex variables theory that the Bergman kernel function of the bounded domains in Cⁿ. Stephen Bergman first introduced the concept of the Bergman kernel in the 1920' s. He used it to study the con formal maps between planar domains. It is just the reproducing kernel of the or-thogonal projection from the space of square-integrable functions to the subspace of square-integrable holomorphic functions. In 1933, Bergman generalized this theory into the case of several complex variables. It is known that every bounded domain admits a Bergman kernel. However, it is difficult to study the Bergman kernel such as the expressions, the zeros or the boundary behavior since few examples except the homogeneous domains and fewer Reinhardt domains of which the Bergman kernel can be computed explicitly. It is also a difficult thing to construct a domain with explicit Bergman kernel. Therefore, some mathematicians considered that domains with explicit Bergman kernels are worthy researching.In 2003, Yin Weiping constructed a new type of domains named Hua construction whose Bergman kernel can be obtained in close form. Similar to the method to Hua domains, we can also compute its Bergman kernel. Firstly, we give the group of holomorphic automorphism of Hua construction, such that the clement F(w, z) of the group maps (w,z) into (w^*,0). According to the transformation formula of theBergman kernel function, we have . It follows that the problem is that we only need compute . Secondly, we introduce the concept of the semi-Reinhardt domain, and compute the complete orthonormal system of it. Because Hua construction are semi-Reinhardt domain, andby the complete orthonormal system, we know is a multi-infinite series about .Then we can get the Bergman kernel function by summing the infinite series and we think we get the explicit formula once the sum is obtained.We take the Hua construction of second type for example.where is the second type symmetric classical domain in the sense of Hua. Z|- denote the conjugate of We have the following result:Chapter IV Zeros of the Bergman kernel-Lu Qikeng conjectureZeros of a Bergman kernel poses an obstruction to the global definition of so-called Bergman representative coordinates. This observation was Lu QiKeng' s motivation for ask the following question: Which domains has zero-free Bergman kernels? It was M. Skwarezynski. a Polish mathematician, who firstly called the above question Lu Qikeng conjecture. A domain of which the Bergman kernel is zero-free is called the Lu QiKeng domain. The explicit expression of the Bergman kernel follows immediately that balls and polydiscs are Lu QiKeng domains. However, to determine whether or not the Bergman kernel of a given domain is quite difficult. It has been conjectured that all strongly pseudoconvex domain with smooth boundary are Lu QiKeng for a long time until H. P. Boas [76] gave a counterexample in 1985. The Bergman kernel of the domain Ω_H= {(z₁,z₂) ∈ C² :} vanishes at the origin for the reason that there is no constant function in the complete orthonormal system of its holomorphic square-integrable functions space. Since it is pseudoconvex. it can be approximated from inside by a sequence of increasing strongly pseudoconvex. complete Reinhardtdomains. By Ramadanov' s theorem and Hurwitz' s theorem, one can immediately obtain the desired counterexample. From then on, counterexamples(i.e. the domains whose Bergman kernel have zeros) arc given almost every year.In this part, we study Lu Qikeng conjecture on the Cartan-Hartogs domain of the first type.Y_I{N,m,n;K) = {W ∈C^N,Z ∈ R_I(m,n) : |W|^2K< det(I- Z(Z|-)^t),K > 0}, We get the following results:· Y_I(N, 1,1; K) is a Lu Qikeng domain;· Y_I(1,1,2; K) is a Lu Qikeng domain if and only if K ≥ 1/2;· Y_I(2, 1, 2; K) is a Lu Qikeng domain if and only if K≥1/4;· Y_I(N, 1,2; K) is always Lu Qikeng domain if N≥ 3;· Y_I(1,1,3;K) is a Lu Qikeng domain if and only if K ≥(2^1/2)/2;· Y_I(N, 1,3;K) is always Lu Qikeng domain if N ≥ 6;· Y_I(1, 1,4: K) is Lu Qikeng domain iff K ≥(6^1/2)/3.