An n-dimensional Hartogs domain DF can be equipped with a natural Kähler metric gF. This paper contains two results. In the first one we prove that if gF is an extremal Kähler metric then (DF , gF ) is holomorphically isometric to an open subset of the n-dimensional complex hyperbolic space. In the second one we prove the same assertion under the assumption that there exists a real holomorphic vector field X on DF such that (gF, X) is a Kähler–Ricci soliton.
Canonical metrics on Hartogs domains
LOI, ANDREA;ZUDDAS, FABIO
2010-01-01
Abstract
An n-dimensional Hartogs domain DF can be equipped with a natural Kähler metric gF. This paper contains two results. In the first one we prove that if gF is an extremal Kähler metric then (DF , gF ) is holomorphically isometric to an open subset of the n-dimensional complex hyperbolic space. In the second one we prove the same assertion under the assumption that there exists a real holomorphic vector field X on DF such that (gF, X) is a Kähler–Ricci soliton.
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