The disc condition and the local maximum principle (Q1105083)

It is shown that for a 1-dimensional parametric family of the Riemann domains over \({\mathbb{C}}^ n\) the disc condition and the plurisubharmonicity of the boundary distant function are equivalent. The realization of the disc condition for open sets constructed from the spectrum of holomorphic operator functions is studied. Further, the equivalence between the disc condition and the extension of holomorphic maps for open sets of complex manifolds satisfying the disc condition is established. Finally the equivalence of the Steinness and the maximum local principal for the Riemann domains over \({\mathbb{C}}^ n\) (n\(\geq 3)\) is proved.