Abstract
We study the completeness of a metric which is related to the Bergman metric of a bounded domain (sometimes called the Burbea metric or Fuks metric). We provide a criterion for its completeness in the spirit of the Kobayashi criterion for the completeness of the Bergman metric. In particular we prove that in hyperconvex domains our metric is complete.
Highlights
Recall that in a bounded domain Ω ⊂⊂ Cn the Bergman metric is the Kähler metric with metric tensor Ti j(z) := ∂2 ∂ zi ∂ zj log K (z, z), z ∈ Ω, i, j =, n, (1)Communicated by A
In this paper we study the completeness of the following Kähler metric
The completeness of Ti jis likewise defined as the property that every Cauchy sequence with respect to distΩ has a limit point in Ω or equivalently that for any z ∈ Ω, z0 ∈ ∂Ω, Ω
Summary
Recall that in a bounded domain Ω ⊂⊂ Cn the Bergman metric is the Kähler metric with metric tensor. The completeness of Ti jis likewise defined as the property that every Cauchy sequence with respect to distΩ has a limit point in Ω or equivalently that for any z ∈ Ω, z0 ∈ ∂Ω, lim ζ →z0 dist Another important property that is shared with the Bergman metric is the fact that domains, which are complete with respect to Ti j, are necessarily pseudoconvex (for the Bergman metric this follows by an old theorem by Bremermann [5], for Ti jthe proof is virtually the same). For this reason we will restrict our attention to bounded pseudoconvex domains in Cn throughout the paper. Ri ci j, which in certain cases may (presumably) be a gain in the study of the completeness of the Bergman metric
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