Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains

Details Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains

2005-05-23

arxiv.org/abs/math/0505469v1

Mathematics

Complex Variables

Scientific paper

Let $D$ be a pseudoconvex domain in $\C^k_t\times\Cn_z$ and let $\phi$ be a plurisubharmonic function in $D$. For each $t$ we consider the $n$-dimensional slice of $D$, $D_t=\{z; (t,z)\in D\}$, let $\phi^t$ be the restriction of $\phi$ to $D_t$ and denote by $K_t(z,\zeta)$ the Bergman kernel of $D_t$ with the weight function $\phi^t$. Generalizing a recent result of Maitani and Yamaguchi (corresponding to $n=1$ and $\phi=0$) we prove that $\log K_t(z,z)$ is a plurisubharmonic function in $D$. We also generalize an earlier results of Yamaguchi concerning the Robin function and discuss similar results in the setting of $\Rn$.

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