A class of nonpositively curved Kähler manifolds biholomorphic to the unit ball in $ {\mathbb{C}}^{n}$

Harish Seshadri; Kaushal Verma

doi:10.1016/j.crma.2006.01.005

Differential Geometry

A class of nonpositively curved Kähler manifolds biholomorphic to the unit ball in

C^{n}

[Une classe des variétés kählériennes, à courbure non positive, holomorphe à une boule dans

C^{n}

]

Présenté par : Étienne Ghys

Harish Seshadri ¹ ; Kaushal Verma ¹

¹ Department of Mathematics, Indian Institute of Science, Bangalore 560012, India

Comptes Rendus. Mathématique, Volume 342 (2006) no. 6, pp. 427-430.

Résumé
Abstract

Soit $(M, g)$ une variété kählérienne complète et simplement connexe à courbure sectionnelle non positive. Supposons que g ait courbure sectionnelle holomorphe constante et négative en delors d'un compact. On démontre que M est biholomorphe à une boule dans $C^{n}$ , où $\dim_{C} M = n$ .

Let $(M, g)$ be a simply connected complete Kähler manifold with nonpositive sectional curvature. Assume that g has constant negative holomorphic sectional curvature outside a compact set. We prove that M is then biholomorphic to the unit ball in $C^{n}$ , where $\dim_{C} M = n$ .

Reçu le : 2005-11-13
Accepté le : 2005-12-20
Publié le : 2006-01-27

DOI : 10.1016/j.crma.2006.01.005

Affiliations des auteurs :

Harish Seshadri ¹ ; Kaushal Verma ¹

¹ Department of Mathematics, Indian Institute of Science, Bangalore 560012, India

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     title = {A class of nonpositively curved {K\"ahler} manifolds biholomorphic to the unit ball in $ {\mathbb{C}}^{n}$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {427--430},
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     language = {en},
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Harish Seshadri; Kaushal Verma. A class of nonpositively curved Kähler manifolds biholomorphic to the unit ball in $ {\mathbb{C}}^{n}$. Comptes Rendus. Mathématique, Volume 342 (2006) no. 6, pp. 427-430. doi : 10.1016/j.crma.2006.01.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.01.005/

Bibliographie
Cité par

[1] W. Ballmann; M. Gromov; V. Schroeder Manifolds of Nonpositive Curvature, Progr. Math., vol. 61, Birkhäuser Boston, Inc., Boston, MA, 1985

[2] J. Bland On the existence of bounded holomorphic functions on complete Kähler manifolds, Invent. Math., Volume 81 (1985), pp. 555-566

[3] D. Burns; S. Shnider Spherical hypersurfaces in complex manifolds, Invent. Math., Volume 33 (1976) no. 3, pp. 223-246

[4] J. Cheeger; D. Ebin Comparison Theorems in Riemannian Geometry, North-Holland Mathematical Library, vol. 9, North-Holland Publishing Co., New York, 1975

[5] R.E. Greene; S.G. Krantz Deformation of complex structures, estimates for the $\bar{\partial}$ equation, and stability of the Bergman kernel, Adv. Math., Volume 43 (1982) no. 1, pp. 1-86

[6] R.E. Greene; H. Wu Gap theorems for noncompact Riemannian manifolds, Duke Math. J., Volume 49 (1982) no. 3, pp. 731-756

[7] Y.T. Siu; S.-T. Yau Complete Kähler manifolds with nonpositive curvature of faster than quadratic decay, Ann. of Math. (2), Volume 105 (1977) no. 2, pp. 225-264

[8] H. Wu Negatively curved Kähler manifolds, Notices Amer. Math. Soc., Volume 14 (1967), p. 515

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