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Comptes Rendus Mathématique
Volume 355, n° 7
pages 760-768 (juillet 2017)
Doi : 10.1016/j.crma.2017.06.009
Received : 2 May 2017 ;  accepted : 28 June 2017
Remarks on the canonical metrics on the Cartan–Hartogs domains
Remarques sur les métriques canoniques des domaines de Cartan–Hartogs
 

Enchao Bi a , Zhenhan Tu b
a School of Mathematics and Statistics, Qingdao University, Qingdao, Shandong 266071, PR China 
b School of Mathematics and Statistics, Wuhan University, Wuhan, Hubei 430072, PR China 

Abstract

The Cartan–Hartogs domains are defined as a class of Hartogs-type domains over irreducible bounded symmetric domains. For a Cartan–Hartogs domain   endowed with the natural Kähler metric  , Zedda conjectured that the coefficient   of the Rawnsley's ε -function expansion for the Cartan–Hartogs domain   is constant on   if and only if   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  .

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Résumé

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   muni de sa métrique de Kähler naturelle  , Zedda a conjecturé que le coefficient   du développement de la fonction ε de Rawnsley relative au domaine de Cartan–Hartogs   est constant sur   si et seulement si   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  .

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