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 ${\Omega }^{B^{d_0}}(\mu )$ endowed with the canonical metric $g(\mu )$, we obtain an explicit formula for the Bergman kernel of the weighted Hilbert space ${\mathcal{H}}_{\alpha }$ of square integrable holomorphic functions on $({\Omega }^{B^{d_0}}(\mu ),g(\mu ))$ with the weight $\exp \{-\alpha \phi \}$ (where $\phi$ is a globally defined Kähler potential for $g(\mu )$) for $\alpha >0$, and, furthermore, we give an explicit expression of the Rawnsley's $\epsilon$-function expansion for $({\Omega }^{B^{d_0}}(\mu ),g(\mu )).$ Secondly, using the explicit expression of the Rawnsley's $\epsilon$-function expansion, we show that the coefficient $a_2$ of the Rawnsley's $\epsilon$-function expansion for the Cartan-Hartogs domain $({\Omega }^{B^{d_0}}(\mu ),g(\mu ))$ is constant on ${\Omega }^{B^{d_0}}(\mu )$ if and only if $({\Omega }^{B^{d_0}}(\mu ),g(\mu ))$ is biholomorphically isometric to the complex hyperbolic space. So we give an affirmative answer to a conjecture raised by M. Zedda.

18 pages, to appear in Mathematische Zeitschrift

Cited by in corpus (2)

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