On D ∗ - Extension Property of the Hartogs Domains

A complex analytic space is said to have the D∗-extension property if and only if any holomorphic map from the punctured disk to the given space extends to a holomorphic map from the whole disk to the same space. A Hartogs domain H over the base X (a complex space) is a subset of X × C where all the fibers over X are disks centered at the origin, possibly of infinite radius. Denote by φ the function giving the logarithm of the reciprocal of the radius of the fibers, so that, when X is pseudoconvex, H is pseudoconvex if and only if φ is plurisubharmonic. We prove that H has the D∗-extension property if and only if (i) X itself has the D∗-extension property, (ii) φ takes only finite values and (iii) φ is plurisubharmonic. This implies the existence of domains which have the D∗-extension property without being (Kobayashi) hyperbolic, and simplifies and generalizes the authors’ previous such example.