On canonical metrics on Cartan–Hartogs domains

Abstract

The Cartan-Hartogs domains are defined as a class of Hartogs type domains over irreducible bounded symmetric domains. The purpose of this paper is twofold. Firstly, for a Cartan-Hartogs domain Ω B d 0 (µ) endowed with the canonical metric g(µ), we obtain an explicit formula for the Bergman kernel of the weighted Hilbert space H α of square integrable holomorphic functions on (Ω B d 0 (µ), g(µ)) with the weight exp{−αϕ} (where ϕ is a globally defined Kähler potential for g(µ)) for α > 0, and, furthermore, we give an explicit expression of the Rawnsley's ε-function expansion for (Ω B d 0 (µ), g(µ)). Secondly, using the explicit expression of the Rawnsley's ε-function expansion, we show that the coefficient a 2 of the Rawnsley's ε-function expansion for the Cartan-Hartogs domain (Ω B d 0 (µ), g(µ)) is constant on Ω B d 0 (µ) if and only if (Ω B d 0 (µ), g(µ)) is biholomorphically isometric to the complex hyperbolic space. So we give an affirmative answer to a conjecture raised by M. Zedda.

Remarks on the canonical metrics on the Cartan–Hartogs domains

“…In 2014, Feng-Tu [8] proved this conjecture by giving the explicit expression of the the Rawnsley's ε-function expansion for the Cartan-Hartogs domain (Ω B (µ), g(µ)). The methods in Feng-Tu [8] are very different from the argument in Zedda [22].…”

“…For the Cartan-Hartogs domain (Ω B (µ), g(µ)), the Rawnsley's ε-function admits the following finite expansion (e.g., see Th. 3.1 in Feng-Tu [8]): …”

“…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)).…”

“…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.…”

Rawnsley’s ε-function on some Hartogs type domains over bounded symmetric domains and its applications

“…If ν = 0, then the metric g(µ; ν) becomes the standard canonical metric (e.g., see Bi-Tu [3], Feng-Tu [14,15], Loi-Zedda [20] and Zedda [27,28]). In this paper, we will focus our attention on the metric…”