Curvature of invariant metrics on Piatetski-Shapiro's generalized homogeneous domains (Q1909234)

The first example of a non-symmetric homogeneous bounded domain was given by Piatetski-Shapiro. In J. Differ. Geom. 15, 61-70 (1980; Zbl 0448.53014) \textit{J. E. D'Atri} presents the original 8-dimensional example of Piatetski-Shapiro and shows that the Bergman metric has planes of positive curvature. Moreover he shows that the Bergman metric can be deformed, through invariant Kähler metrics, to a metric with non-positive sectional curvature. He points out that it would be of interest to know the full isometry group of this curve of metrics. In the present article the author studies, in a class of solvable Lie algebras of dimension \(6 + 2p\), \(p > 0\) (a generalization of the 8-dimensional example of Piatetski-Shapiro) all invariant Kähler metrics of non-positive sectional (holomorphic sectional) curvature. It is shown that they are obtained by a one-parameter family of (non-isometric) invariant, non-symmetric Kähler metrics. Furthermore the corresponding groups of isometries are determined. It is shown that they all coincide.