Explicit formulas of the Bergman kernel for 3-dimensional complex ellipsoids
The author considers the complex ellipsoids \[ \Omega _{p_1, \dots , p_n} = \bigg\{ (z_1, \dots , z_n) \in \mathbb C^n : \sum _{j=1}^n |z_j|^{2p_j} < 1 \bigg\} \] where each \(p_j\) is a positive real number. The aim is to obtain an explicit formula for the Bergman kernel for such ellipsoids in closed form, i.e., without writing down an infinite series. Because of difficulties, results of this type were obtained up to now by several authors only for a rather limited set of domains \(\Omega _{p_1, \dots , p_n}\). The main theorems of the present paper are: Theorem 1.1. The Bergman kernel for \(\big\{ (z_1, z_2, z_3) \in \mathbb C^3: |z_1|^4+ |z_2|^4+ |z_3|^4 <1 \big\}\) is given by \[ \begin{multlined} K(z,w) = \frac {48 \eta _1 \eta _2 \eta _3}{\pi ^3(1- \eta _1^2 - \eta _2^2 - \eta _3^2)^4} + \frac {48 (1+\eta _1 +\eta _2 +\eta _3 )}{\pi ^4(1- \eta _1^2 - \eta _2^2 - \eta _3^2)^3}\\ +\frac 6{\pi ^4}\diamondsuit _3 \bigg\{ \sum _{(i,j,k)\in J} \frac {c'_{ijk}(1+\eta _1 + \eta _2)^{1-k}(1+\eta _1)^k} {(1-\eta _1^2 -\eta _2^2 -\eta _3^2 )^i(1-\eta _1^2 -\eta _2^2 )^j (1-\eta _1^2)^k} \bigg\} \\ + \frac {3}{\pi ^4 (1-\eta _1^2 - \eta _2^2 - \eta _3^2 )^4 } \diamondsuit _3 \bigg( \frac {\eta _1 \eta _2 \sum _{i+j=3}c_{ij}(1-\eta _1 ^2 - \eta _2 ^2 )^i (\eta _3^2)^j} {(1-\eta _1^2-\eta _2^2)^{\frac 52}} \\ \times \bigg( \pi + 2\arcsin \frac {\eta _1}{\sqrt{1-\eta _2^2}} +2 \arcsin \frac {\eta _2} {\sqrt{1-\eta _1^2}} +2 \arcsin \frac {\eta _1 \eta _2}{\sqrt{1-\eta _1^2}\sqrt{1-\eta _2^2}}\bigg)\bigg) \end{multlined} \] where \(\eta _j = z_j\overline w_j\) for \(j=1,2,3\), \(\diamondsuit _3\) is defined by \[ \diamondsuit _3 f(\eta _1, \eta _2, \eta _3) = f(\eta _1, \eta _2, \eta _3) + f(\eta _2, \eta _3, \eta _1) + f(\eta _3, \eta _1, \eta _2), \] \(J = \{ (2,1,0), (1,2,0), (1,1,1) \} \) and \(c'_{210}= -2\), \(c'_{111} = \frac 13 \), \(c_{30}=5\), \(c_{21}=15\), \(c_{12}=-5\), \(c_{03}=1\). Then the author states Theorem 1.2, in which a closed form of the Bergman kernel for the domain \(\big\{ (z_1, z_2, z_3) \in \mathbb C^3: |z_1|^4+ |z_2|^4+ |z_3|^2 <1 \big\}\) is presented. As the author says, the above domains are the first examples of 3-dimensional complex ellipsoids with their Bergman kernel written in a closed form, i.e., avoiding infinite series. To prove Theorems 1.1 and 1.2, the author computes explicit forms of the Appel hypergeometric functions \[ F_A^{(n)} \bigg(\alpha; 1, \dots , 1; \frac 12, \dots , \frac 12; \zeta _1, \dots , \zeta _n\bigg), \] with \(\alpha = \frac 12\), \(n=1,2,3\), and where \[ \begin{multlined} F_A^{(n)} \big(\alpha; \beta _1, \dots , \beta _n; \gamma _1, \dots , \gamma _n; \zeta _1, \dots , \zeta _n\big)\\ = \sum _{m_1=0}^\infty \dots \sum _{m_n=0}^\infty \frac {(\alpha )_{m_1+ \dots +m_n}}{m_1! \dots m_n! } \frac {(\beta _1)_{m_1}\dots (\beta _n)_{m_n}}{(\gamma _1)_{m_1}\dots (\gamma _n)_{m_n}} \zeta _1^{m_1} \dots \zeta _n^{m_n} . \end{multlined} \] Then the author obtains some recursion formula by means of which one can reduce the computation of higher dimensional Appel hypergeometric functions to the computation of one or two dimensional Appel hypergeometric functions. By means of that recursion formula, the author obtains explicit formulas for the Bergman kernel of the complex ellipsoids \(\Omega _{2,2,2}\) and \(\Omega _{2,2,1}\) in the forms announced in Theorems 1.1 and 1.2. The last chapter of the paper is devoted to the study of the behaviour of the Bergman kernel near weakly pseudoconvex boundary points of \(\Omega _{2,2,2}\) and \(\Omega _{2,2,1}\) by means of the just obtained formulas for the Bergman kernels of those domains.
scientific article; zbMATH DE number 6143243