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 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.
@article{urn:eudml:doc:41436,
title = {On D*-extension property of the Hartogs domains.},
journal = {Publicacions Matem\`atiques},
volume = {45},
year = {2001},
pages = {421-429},
mrnumber = {MR1876915},
zbl = {0994.32014},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41436}
}
Thai, Do Duc; Thomas, Pascal J. On D*-extension property of the Hartogs domains.. Publicacions Matemàtiques, Tome 45 (2001) pp. 421-429. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41436/