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Comptes Rendus Mathématique
Volume 335, n° 9
pages 751-756 (novembre 2002)
Doi : S1631-073X(02)02558-X
Received : 18 July 2002 ;  accepted : 12 September 2002
Courants de type Liouville pour les applications holomorphes
Liouville type currents for holomorphic maps
 

Said Asserda a , M'hamed Kassi b
a Rue 326, 51 Kénitra, Maroc 
b Département de mathématiques, Université Mohamed-V, BP 1014 Rabat, Maroc 

Note présentée par Jean-Pierre Demailly

Résumé

On montre que tout courant positif fermé T , régularisable et à croissance faible dans une variété Kählerienne M est de Liouville relatif à la classe des applications holomorphes bornées sur le support de T et à valeurs dans une variété Kählerienne N de forme de Kähler exacte. On établit un théorème de type Casorati-Weierstrass pour le courant T . Aussi, on montre que si (M ,ω ) est Kählerienne complète de courbure de Ricci semi-positive à l'infini i.e. Ricω (x )⩾− (r (x )) où (t ) decroit vers 0 à l'infini, alors M est de Liouville pourvu qu'il existe p >1 tel que λ 1 (M )⩾p (0) et la fonction max( (r ),r −2) est p -sommable à l'infini. Pour citer cet article : S. Asserda, M. Kassi, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 751-756.

The full text of this article is available in PDF format.
Abstract

We show that every regularized positif closed current T with slow growth on a Kähler manifold M is a Liouville current with respect to the class of holomorphic maps bounded on the support of T with values on a Kähler manifold N whose Kähler form is exact. We establish a Casorati-Weierstrass type theorem for the current T . Also we show that if (M ,ω ) is a complete Kähler manifold with nonnegative Ricci curvature at infinity, i.e., Ricω (x )⩾− (r (x )) where (t ) is nonnegative and decreass to 0 at infinity, then M is a Liouville manifold provided that λ 1 (M )⩾p (0) and the function max( (r ),r −2) is p -summable at infinity for some p >1. To cite this article: S. Asserda, M. Kassi, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 751-756.

The full text of this article is available in PDF format.


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