Maximal plurisubharmonic functions and the polynomial hull of a completely circled fibration

Abstract

LetX(-ϱB^m×Cⁿ be a compact set over the unit sphere ϱB^m such that for each z∈ϱB^m the fiber X_z={ω∈Cⁿ;(z, ω)∈X} is the closure of a completely circled pseudoconvex domain in Cⁿ. The polynomial hull $\hat X$ of X is described in terms of the Perron-Bremermann function for the homogeneous defining function of X. Moreover, for each point (z₀, w₀)∈Int $\hat X$ there exists a smooth up to the boundary analytic disc F:Δ→B^m×Cⁿ with the boundary in X such that F(0)=(z₀, w₀).