On $\mathbb{D}^*$-extension property of the Hartogs domains

A complex analytic space is said to have the $\mathbb{D}^*$-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 $X \times \mathbb{C}$ where all the fibers over $X$ are disks centered at the origin, possibly of infinite radius. Denote by $\phi$ 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 $\phi$ is plurisubharmonic.

We prove that $H$ has the $\mathbb{D}^*$-extension property if and only if (i) $X$ itself has the $\mathbb{D}^*$-extension property, (ii) $\phi$ takes only finite values and (iii) $\phi$ is plurisubharmonic. This implies the existence of domains which have the $\mathbb{D}^*$-extension property without being (Kobayashi) hyperbolic, and simplifies and generalizes the authors' previous such example.