Liouville-type theorems for foliations with complex leaves
[Théorèmes du type Liouville pour feuilletages avec feuilles complexes]
Annales de l'Institut Fourier, Tome 60 (2010) no. 2, pp. 711-725.

Dans cet article nous considérons différentes questions relatives à la structure du feuilletage de certaines sous-variétés S n , en particulier les variétés Levi-plates. Comme schéma général, on suppose que S est bornée le long d’une coordonnée (ou d’un sous-ensemble des coordonnées), et on montre que les feuilles complexes de son feuilletage sont des plans.

In this paper we discuss various problems regarding the structure of the foliation of some foliated submanifolds S of n , in particular Levi flat ones. As a general scheme, we suppose that S is bounded along a coordinate (or a subset of coordinates), and prove that the complex leaves of its foliation are planes.

DOI : 10.5802/aif.2537
Classification : 32V40
Keywords: Levi flat submanifolds, Liouville theorem, analytic multifunctions
Mot clés : sous-variétés Levi-plates, Théorème de Liouville, multifonctions analytiques
Della Sala, Giuseppe 1

1 Institut fuer Mathematik Nordbergstrasse 15 1090 Wien (Autriche)
@article{AIF_2010__60_2_711_0,
     author = {Della Sala, Giuseppe},
     title = {Liouville-type theorems for foliations with complex leaves},
     journal = {Annales de l'Institut Fourier},
     pages = {711--725},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {60},
     number = {2},
     year = {2010},
     doi = {10.5802/aif.2537},
     zbl = {1194.32026},
     mrnumber = {2667791},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2537/}
}
TY  - JOUR
AU  - Della Sala, Giuseppe
TI  - Liouville-type theorems for foliations with complex leaves
JO  - Annales de l'Institut Fourier
PY  - 2010
SP  - 711
EP  - 725
VL  - 60
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2537/
DO  - 10.5802/aif.2537
LA  - en
ID  - AIF_2010__60_2_711_0
ER  - 
%0 Journal Article
%A Della Sala, Giuseppe
%T Liouville-type theorems for foliations with complex leaves
%J Annales de l'Institut Fourier
%D 2010
%P 711-725
%V 60
%N 2
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2537/
%R 10.5802/aif.2537
%G en
%F AIF_2010__60_2_711_0
Della Sala, Giuseppe. Liouville-type theorems for foliations with complex leaves. Annales de l'Institut Fourier, Tome 60 (2010) no. 2, pp. 711-725. doi : 10.5802/aif.2537. http://www.numdam.org/articles/10.5802/aif.2537/

[1] Barrett, D. E.; Fornæss, J. E. On the smoothness of Levi-foliations, Publ. Mat., Volume 32 (1988) no. 2, pp. 171-177 | MR | Zbl

[2] Barrett, David E.; Inaba, Takashi On the topology of compact smooth three-dimensional Levi-flat hypersurfaces, J. Geom. Anal., Volume 2 (1992) no. 6, pp. 489-497 | MR | Zbl

[3] Bryant, Robert L. Levi-flat minimal hypersurfaces in two-dimensional complex space forms, Lie groups, geometric structures and differential equations—one hundred years after Sophus Lie (Kyoto/Nara, 1999) (Adv. Stud. Pure Math.), Volume 37, Math. Soc. Japan, Tokyo, 2002, pp. 1-44 | MR | Zbl

[4] Fornaess, John Erik; Stout, Edgar Lee Polydiscs in complex manifolds, Math. Ann., Volume 227 (1977) no. 2, pp. 145-153 | DOI | MR | Zbl

[5] Lins Neto, Alcides A note on projective Levi flats and minimal sets of algebraic foliations, Ann. Inst. Fourier (Grenoble), Volume 49 (1999) no. 4, pp. 1369-1385 | DOI | Numdam | MR | Zbl

[6] Lupacciolu, Guido; Tomassini, Giuseppe An extension theorem for CR-functions, Ann. Mat. Pura Appl. (4), Volume 137 (1984), pp. 257-263 | DOI | MR | Zbl

[7] Ohsawa, Takeo On the Levi-flats in complex tori of dimension two, Publ. Res. Inst. Math. Sci., Volume 42 (2006) no. 2, pp. 361-377 | DOI | MR | Zbl

[8] Oka, Kiyosi Note sur les familles de fonctions analytiques multiformes, J. Sci. Hiroshima Univ. Ser. A, Volume 4 (1934), pp. 93-98 | Zbl

[9] Pugh, Charles C. The Morse-Sard theorem with mixed differentiability, Indiana Univ. Math. J., Volume 40 (1991) no. 4, pp. 1387-1396 | DOI | MR | Zbl

[10] Ransford, Thomas A new approach to analytic multifunctions, Set-Valued Anal., Volume 7 (1999) no. 2, pp. 159-194 | DOI | MR | Zbl

[11] Shcherbina, N. V. On the polynomial hull of a graph, Indiana Univ. Math. J., Volume 42 (1993) no. 2, pp. 477-503 | DOI | MR | Zbl

[12] Siu, Yum-Tong Nonexistence of smooth Levi-flat hypersurfaces in complex projective spaces of dimension 3, Ann. of Math. (2), Volume 151 (2000) no. 3, pp. 1217-1243 | DOI | MR | Zbl

[13] Slodkowski, Zbigniew Analytic set-valued functions and spectra, Math. Ann., Volume 256 (1981) no. 3, pp. 363-386 | DOI | MR | Zbl

[14] Stolzenberg, Gabriel Uniform approximation on smooth curves, Acta Math., Volume 115 (1966), pp. 185-198 | DOI | MR | Zbl

Cité par Sources :