Compressed Product of Balls and Lower Boundary Estimates on Bergman Kernels

Abstract

The image $B_{p^{\sigma},q}$ of a product of balls $B_p \times B_q$ under a compression $c_{\sigma}(X,V) = (X,V(1 - ^t(\bar{X}X)^{\frac{\sigma}{2})}$ is called acompressed product of balls of exponent $\sigma \in \mathbb{R}$. The present note obtains the group Aut($B_{p^{\sigma},q}$) of the holomorphic automorphisms and the Aut($B_{p^{\sigma},q}$)-orbit structure of $B_{p^{\sigma},q}$ and its boundary ${\partial}B_{p^{\sigma},q}$ for $\sigma \gt 1$. The Bergman completeness of $B_{p^{\sigma},q}$ is verified by an explicit calculation of the Bergman kernel. As a consequence, local lower boundary estimates on the Bergman kernels of the bounded pseudoconvex domains are obtained, which are locally inscribed in $B_{p^{\sigma},q}$ at a common boundary point.