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An n-dimensional Hartogs domain D F can be equipped with a natural Kähler metric g F . This paper contains two results. In the first one we prove that if g F is an extremal Kähler metric then (D F , g F ) 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 D F such that (g F , X) is a Kähler-Ricci soliton.doi:10.18910/8607 fatcat:itimrjzwj5cuvdbi4gxcgk6ca4