Some Topics Related to Bergman Kernel
Actually we will discuss some topics related to Bergman kernel on Cartan- Hartogs domain.
math.CV↗
arXiv subjects
Weiping Yin
Publications and source records attributed to Weiping Yin.
Lu Qi-Keng Conjectue and Hua Domain
The first part I talk about the motivation for Lu Qi-Keng conjecture and the results about the presence or absence of zeroes of the Bergman kernel function of a bounded domain in ${\bf{C^n}}$. The second part I summarize the main results on Hua domains, such as the explicit Bergman kernel function, comparison theorem for the invariant metrics, explicit complete Einstein-Kähler metrics, the equivalence between the Einstein-Kähler metric and the Bergman metric etc.
math.CV↗
Two Problems on Cartan Domains
Firstly, we consider the unitary geometry of two exceptional Cartan domains $\Re_{V}(16)$ and $\Re_{VI}(27)$. We obtain the explicit formulas of Bergman kernal funtion, Cauchy-Szegö kernel, Poinsson kernel and Bergman metric for $\Re_{V}(16)$ and $\Re_{VI}(27)$. Secondly, we give a class of invariant differential operators for Cartan domain $\Re$ of dimension n: If the Bergman metric of $\Re$ is $$ds^{2}=\sum\limits_{i,j=1}^{n}g_{ij}dz_{i}d\bar{z}_{j}, T(z,\bar{z})=(g_{ij})$$ and $$L(u)=T^{-1}(z,\bar{z}) [\frac{\partial^2u}{\partial z_i\partial\bar{z}_j}],$$then $$L_j(u)=\{\mbox {The sum of all prinipal minors of degree} j {for} L(u)\}$$ is invariant under the biholomorphic mapping of $\Re$. Let $D$ be the irreducible bounded homogeneous domain in $C^n$, $P=P(z,*)$ the Poisson kernel of $D$, then for any fixed $J(1\leq j \leq n)$ one has $L_j(P^{1/j})=0$ iff $D$ is a symmetric domain.
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The equivalent classical metrics on the Cartan-Hartogs Domains
In this paper we study the complete invariant metrics on Cartan-Hartogs 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.
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Complete Einstein-Kähler Metric and Holomorphic Sectional Curvature on $Y_{II}(r,p;K)$
The explicit complete Einstein-Kähler metric on the second type Cartan-Hartogs domain $Y_{II}(r,p;K)$ is obtained in this paper when the parameter $K$ equals $\frac p2+\frac 1{p+1}$. The estimate of holomorphic sectional curvature under this metric is also given which intervenes between $-2K$ and $-\frac{2K}p$ and it is a sharp estimate. In the meantime we also prove that the complete Einstein-Kähler metric is equivalent to the Bergman metric on $Y_{II}(r,p;K)$ when $K=\frac p2+\frac 1{p+1}$.
math.CV↗
New classes of domains with explicit Bergman kernel
We introduce two classes of "egg type" domains, built on general bounded symmetric domains, for which we compute the Bergmann kernel in explicit form. We use the characterization of bounded symmetric domains through Jordan triple systems. The egg type domains are defined using the generic norm. A generalization of the Hua integral is computed; the result shows the existence of a special polynomial with integer or half-integer coefficients, attached to each irreducible bounded symmetric domain.
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