Canonical Metrics on Cartan–hartogs Domains

Abstract

In this paper we address two problems concerning a family of domains M Ω (µ) ⊂ C n , called Cartan-Hartogs domains, endowed with a natural Kähler metric g(µ). The first one is determining when the metric g(µ) is extremal (in the sense of Calabi), while the second one studies when the coefficient a 2 in the Engliš expansion of Rawnsley ε-function associated to g(µ) is constant.

Remarks on the canonical metrics on the Cartan–Hartogs domains

“…Lemma 2.1 (Zedda [22], Lemma 4). The scalar curvature k g(µ) of the Cartan-Hartogs domain (Ω B (µ), g(µ)) is given by…”

“…Recently, Loi-Zedda [11] and Zedda [22] studied the canonical metrics on the CartanHartogs domains. By calculating the scalar curvature k g(µ) , the Laplace ∆k g(µ) of k g(µ) , the norm |R g(µ) | 2 of the curvature tensor R g(µ) and the norm |Ric g(µ) | 2 of the Ricci curvature Ric g(µ) of a Cartan-Hartogs domain (Ω B d 0 (µ), g(µ)), Zedda [22] has proved that if the coefficient a 2 of the Rawnsley's ε-function expansion for the Cartan-Hartogs domain (Ω B (µ), g(µ)) is constant on Ω B (µ), then (Ω B (µ), g(µ)) is Kähler-Einstein.…”

“…By calculating the scalar curvature k g(µ) , the Laplace ∆k g(µ) of k g(µ) , the norm |R g(µ) | 2 of the curvature tensor R g(µ) and the norm |Ric g(µ) | 2 of the Ricci curvature Ric g(µ) of a Cartan-Hartogs domain (Ω B d 0 (µ), g(µ)), Zedda [22] has proved that if the coefficient a 2 of the Rawnsley's ε-function expansion for the Cartan-Hartogs domain (Ω B (µ), g(µ)) is constant on Ω B (µ), then (Ω B (µ), g(µ)) is Kähler-Einstein. Moreover, Zedda [22] conjectured that the coefficient a 2 of the Rawnsley's ε-function expansion for the Cartan-Hartogs domain (Ω B (µ), g(µ)) is constant on Ω B (µ) if and only if (Ω B (µ), g(µ)) is biholomorphically isometric to the complex hyperbolic space.…”

“…The methods in Feng-Tu [8] are very different from the argument in Zedda [22]. In this paper, following the framework of Zedda [22], we give a geometric proof of the Zedda's conjecture by computing the curvature tensors of the Cartan-Hartogs domain (Ω B (µ), g(µ)). We will prove the following result:…”

“…As general references for this paper, see Feng-Tu [8] and Zedda [22]. For the sake of simplicity, simliar to Zedda [22], the Cartan-Hatogs domains in this paper will be restricted to the Hartogs type domains over the irreducible bounded symmetric domains only with one-dimensional fibers.…”