no. 6
Differential Geometry
A class of nonpositively curved Kähler manifolds biholomorphic to the unit ball in Cn
[Une classe des variétés kählériennes, à courbure non positive, holomorphe à une boule dans Cn]

Présenté par : Étienne Ghys

Seshadri, Harish1 ; Verma, Kaushal1
1 Department of Mathematics, Indian Institute of Science, Bangalore 560012, India
Comptes Rendus. Mathématique, Tome 342 (2006) no. 6, pp. 427-430.

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 Cn, où dimCM=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 Cn, where dimCM=n.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2006.01.005
Seshadri, Harish 1 ; Verma, Kaushal 1

1 Department of Mathematics, Indian Institute of Science, Bangalore 560012, India 
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Seshadri, Harish; Verma, Kaushal. A class of nonpositively curved Kähler manifolds biholomorphic to the unit ball in $ {\mathbb{C}}^{n}$. Comptes Rendus. Mathématique, Tome 342 (2006) no. 6, pp. 427-430. doi : 10.1016/j.crma.2006.01.005. http://www.numdam.org/articles/10.1016/j.crma.2006.01.005/

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

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

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

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

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

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

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

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

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