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Canonical metrics on Hartogs domains. (English) Zbl 1484.53105

Summary: 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.

MSC:

53C55 Global differential geometry of Hermitian and Kählerian manifolds
32Q15 Kähler manifolds
32T15 Strongly pseudoconvex domains
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References:

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