Remarks on the canonical metrics on the Cartan–Hartogs domains

Enchao Bi; Zhenhan Tu

doi:10.1016/j.crma.2017.06.009

Complex analysis/Differential geometry

Remarks on the canonical metrics on the Cartan–Hartogs domains
[Remarques sur les métriques canoniques des domaines de Cartan–Hartogs]

Présenté par : the Editorial Board

Enchao Bi ¹ ; Zhenhan Tu ²

¹ School of Mathematics and Statistics, Qingdao University, Qingdao, Shandong 266071, PR China
² School of Mathematics and Statistics, Wuhan University, Wuhan, Hubei 430072, PR China

Comptes Rendus. Mathématique, Volume 355 (2017) no. 7, pp. 760-768.

Abstract (VO)
Résumé

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 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. 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 $(Ω^{B} (μ), g (μ))$ .

Les domaines de Cartan–Hartogs sont définis comme une classe de domaines de type Hartogs sur les domaines symétriques bornés irréductibles. Pour un domaine de Cartan–Hartogs $Ω^{B} (μ)$ muni de sa métrique de Kähler naturelle $g (μ)$ , Zedda a conjecturé que le coefficient $a_{2}$ du développement de la fonction ε de Rawnsley relative au domaine de Cartan–Hartogs $(Ω^{B} (μ), g (μ))$ est constant sur $Ω^{B} (μ)$ si et seulement si $(Ω^{B} (μ), g (μ))$ est biholomorphiquement isométrique à l'espace hyperbolique complexe. Dans cet article, en nous appuyant sur ses arguments, nous donnons une preuve géométrique de la conjecture de Zedda en calculant les tenseurs de courbure du domaine de Cartan–Hartogs $(Ω^{B} (μ), g (μ))$ .

Reçu le : 2017-05-02
Accepté le : 2017-06-28
Publié le : 2017-07-01

DOI : 10.1016/j.crma.2017.06.009

Affiliations des auteurs :

Enchao Bi ¹ ; Zhenhan Tu ²

¹ School of Mathematics and Statistics, Qingdao University, Qingdao, Shandong 266071, PR China
² School of Mathematics and Statistics, Wuhan University, Wuhan, Hubei 430072, PR China

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     author = {Enchao Bi and Zhenhan Tu},
     title = {Remarks on the canonical metrics on the {Cartan{\textendash}Hartogs} domains},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {760--768},
     publisher = {Elsevier},
     volume = {355},
     number = {7},
     year = {2017},
     doi = {10.1016/j.crma.2017.06.009},
     language = {en},
}

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Enchao Bi; Zhenhan Tu. Remarks on the canonical metrics on the Cartan–Hartogs domains. Comptes Rendus. Mathématique, Volume 355 (2017) no. 7, pp. 760-768. doi : 10.1016/j.crma.2017.06.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.06.009/

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