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arXiv · 1706.09775

Remarks on the canonical metrics on the Cartan-Hartogs domains

Enchao Bi·Zhenhan Tu
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Abstract

The Cartan-Hartogs domains are defined as a class of Hartogs type domains over irreducible bounded symmetric domains. For a Cartan-Hartogs domain $Ω^{B}(μ)$ endowed with the natural Kähler metric $g(μ),$ Zedda conjectured that the coefficient $a_2$ of the Rawnsley's $\varepsilon$-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. In this paper, following Zedda's argument, we give a geometric proof of the Zedda's conjecture by computing the curvature tensors of the Cartan-Hartogs domain $(Ω^{B}(μ), g(μ))$.

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Enchao Bi, Zhenhan Tu. 2017-06-29. Remarks on the canonical metrics on the Cartan-Hartogs domains. https://arxiv.org/abs/1706.09775

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