Complete real Kähler Euclidean hypersurfaces are cylinders
[Toute hypersurface Euclidienne réelle Kahlerienne et complète est un cylindre]
Annales de l'Institut Fourier, Tome 57 (2007) no. 1, pp. 155-161.

Dans cet article nous montrons que toute hypersurface Kählerienne complète immergée dans un espace Euclidien M 2n 2n+1 est le produit d’une surface de 3 et d’un facteur Euclidien n-1 2n-2 .

In this note we show that any complete Kähler (immersed) Euclidean hypersurface M 2n 2n+1 must be the product of a surface in 3 with an Euclidean factor n-1 2n-2 .

DOI : 10.5802/aif.2254
Classification : 53C40, 53C55
Keywords: Kähler submanifolds, cylinders, splitting
Mot clés : Kähler hypersurface, cylindres, fendre
Florit, Luis A. 1 ; Zheng, Fangyang 2

1 IMPA: Estrada Dona Castorina 110 22460–320, Rio de Janeiro (Brazil)
2 Ohio State University Columbus, OH 43210 (USA) and Zhejiang University IMS Hanzhou (China)
@article{AIF_2007__57_1_155_0,
     author = {Florit, Luis A. and Zheng, Fangyang},
     title = {Complete real {K\"ahler} {Euclidean} hypersurfaces are cylinders},
     journal = {Annales de l'Institut Fourier},
     pages = {155--161},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {57},
     number = {1},
     year = {2007},
     doi = {10.5802/aif.2254},
     zbl = {1119.53005},
     mrnumber = {2313088},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2254/}
}
TY  - JOUR
AU  - Florit, Luis A.
AU  - Zheng, Fangyang
TI  - Complete real Kähler Euclidean hypersurfaces are cylinders
JO  - Annales de l'Institut Fourier
PY  - 2007
SP  - 155
EP  - 161
VL  - 57
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2254/
DO  - 10.5802/aif.2254
LA  - en
ID  - AIF_2007__57_1_155_0
ER  - 
%0 Journal Article
%A Florit, Luis A.
%A Zheng, Fangyang
%T Complete real Kähler Euclidean hypersurfaces are cylinders
%J Annales de l'Institut Fourier
%D 2007
%P 155-161
%V 57
%N 1
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.2254/
%R 10.5802/aif.2254
%G en
%F AIF_2007__57_1_155_0
Florit, Luis A.; Zheng, Fangyang. Complete real Kähler Euclidean hypersurfaces are cylinders. Annales de l'Institut Fourier, Tome 57 (2007) no. 1, pp. 155-161. doi : 10.5802/aif.2254. http://archive.numdam.org/articles/10.5802/aif.2254/

[1] Abe, K. A complex analogue of Hartman-Nirenberg cylinder theorem, J. Differential Geom., Volume 7 (1972), pp. 453-460 | MR | Zbl

[2] Abe, K. On a class of hypersurfaces of 2n+1 , Duke Math. J., Volume 41 (1974), pp. 865-874 | DOI | MR | Zbl

[3] Dajczer, M.; Gromoll, D. Real Kähler submanifolds and uniqueness of the Gauss map, J. Differential Geom., Volume 22 (1985), pp. 13-28 | MR | Zbl

[4] Dajczer, M.; Gromoll, D. Rigidity of complete Euclidean hypersurfaces, J. Differential Geom., Volume 31 (1990), pp. 401-416 | MR | Zbl

[5] Dajczer, M.; Rodríguez, L. Complete real Kähler minimal submanifolds, J. Reine Angew. Math., Volume 419 (1991), pp. 1-8 | MR | Zbl

[6] Florit, L.; Hui, W.; Zheng, F. On real Kähler Euclidean submanifolds with non-negative Ricci curvature, J. Eur. Math. Soc., Volume 7 (2005), pp. 1-11 | DOI | MR | Zbl

[7] Florit, L.; Zheng, F. Complete real Kähler Euclidean submanifolds in codimension two (Preprint at http://www.preprint.impa.br/Shadows/SERIE_A/2004/306.html)

[8] Florit, L.; Zheng, F. A local and global splitting result for real Kähler Euclidean submanifolds, Arch. Math. (Basel), Volume 84 (2005), pp. 88-95 | MR | Zbl

[9] Hartman, P. On isometric immersions in Euclidean space of manifolds with non–negative sectional curvatures II, Trans. Amer. Math. Soc., Volume 147 (1970), pp. 529-540 | DOI | Zbl

[10] Hartman, P.; Nirenberg, L. On spherical image maps whose Jacobians do not change sign, Amer. J. Math., Volume 81 (1959), pp. 901-920 | DOI | MR | Zbl

[11] Ryan, P. Kähler manifolds as real hypersurfaces, Duke Math. J., Volume 40 (1973), pp. 207-213 | DOI | MR | Zbl

[12] Takahashi, T. A note on Kählerian hypersurfaces of spaces of constant curvature, Kumamoto J. Sci. (Math.), Volume 9 (1972), pp. 21-24 | Zbl

Cité par Sources :