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

Presented by: 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
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 $(Ω^{B} (μ), g (μ))$ .

Received: 2017-05-02
Accepted: 2017-06-28
Published online: 2017-07-01

DOI: 10.1016/j.crma.2017.06.009

Author's affiliations:

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/

References
Cited by

[1] A. Borel On the curvature tensor of the Hermitian symmetric manifolds, Ann. of Math. (2), Volume 71 (1960) no. 3, pp. 508-521

[2] M. Cahen; S. Gutt; J. Rawnsley Quantization of Kähler manifolds. I: geometric interpretation of Berezin's quantization, J. Geom. Phys., Volume 7 (1990), pp. 45-62

[3] E. Calabi; E. Vesentini On compact, locally symmetric Kähler manifolds, Ann. of Math., Volume 71 (1960) no. 3, pp. 472-507

[4] D. Catlin The Bergman kernel and a theorem of Tian, Katata, 1997 (Trends in Mathematics), Birkhäuser Boston, Boston, MA (1999), pp. 1-23

[5] S. Donaldson Scalar curvature and projective embeddings, I, J. Differ. Geom., Volume 59 (2001), pp. 479-522

[6] M. Engliš A Forelli–Rudin construction and asymptotics of weighted Bergman kernels, J. Funct. Anal., Volume 177 (2000) no. 2, pp. 257-281

[7] M. Engliš The asymptotics of a Laplace integral on a Kähler manifold, J. Reine Angew. Math., Volume 528 (2000), pp. 1-39

[8] Z. Feng; Z. Tu On canonical metrics on Cartan–Hartogs domains, Math. Z., Volume 278 (2014) no. 1, pp. 301-320

[9] S. Helgason Differential Geometry, Lie groups, and Symmetric Spaces, Academic Press, 1979

[10] A. Korányi Analytic invariants of bounded symmetric domains, Proc. Amer. Math. Soc., Volume 19 (1968) no. 2, pp. 279-284

[11] A. Loi; M. Zedda Balanced metrics on Cartan and Cartan–Hartogs domains, Math. Z., Volume 270 (2012), pp. 1077-1087

[12] Z. Lu On the lower order terms of the asymptotic expansion of Tian–Yau–Zelditch, Amer. J. Math., Volume 122 (2000) no. 2, pp. 235-273

[13] X. Ma; G. Marinescu Holomorphic Morse Inequalities and Bergman Kernels, Progress in Mathematics, vol. 254, Birkhäuser Boston Inc., Boston, MA, USA, 2007

[14] X. Ma; G. Marinescu Generalized Bergman kernels on symplectic manifolds, Adv. Math., Volume 217 (2008) no. 4, pp. 1756-1815

[15] X. Ma; G. Marinescu Berezin–Toeplitz quantization on Kähler manifolds, J. Reine Angew. Math., Volume 662 (2012), pp. 1-56

[16] N. Mok Metric Rigidity Theorems on Hermitian Locally Symmetric Manifolds, World Scientific, 1989

[17] J. Rawnsley Coherent states and Kähler manifolds, Q. J. Math., Volume 28 (1977) no. 2, pp. 403-415

[18] Y.-T. Siu Strong rigidity of compact quotients of exceptional bounded symmetric domains, Duke Math. J., Volume 48 (1981) no. 4, pp. 857-871

[19] V.S. Varadarajan Lie Groups, Lie Algebras, and Their Representation, Springer, 1984

[20] H. Xu A closed formula for the asymptotic expansion of the Bergman kernel, Commun. Math. Phys., Volume 314 (2012), pp. 555-585

[21] W.P. Yin; A. Wang The equivalence on classical metrics, Sci. China Ser. A, Math., Volume 50 (2007) no. 2, pp. 183-200

[22] M. Zedda Canonical metrics on Cartan–Hartogs domains, Int. J. Geom. Methods Mod. Phys., Volume 09 (2012) no. 1

[23] S. Zelditch Szegö kernels and a theorem of Tian, Int. Math. Res. Not., Volume 6 (1998), pp. 317-331

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