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Abstract. Let X be a complex space and H a positive homogeneous plurisubharmonic function H on X × C m . Consider the Hartogs-type domain Ω H (X) := {(z, w) ∈ X × C m : H(z, w) < 1}. Let S be an analytic subset of X. We give necessary and sufficient conditions for hyperbolicity and tautness modulo S × C m of Ω H (X), with the obvious corollaries for the special case of Hartogs domains.

Abstract. In this note, we consider models in C 2 . The purpose of this note is twofold. We first show a characterization of models in C 2 by their noncompact automorphism groups. Then we give an explicit description for automorphism groups of models in C 2 .