Closed hypersurfaces of $ {\mathbb{S}}^{4}(1)$ with constant mean curvature and zero Gauß–Kronecker curvature

Lusala, Tsasa; Gomes de Oliveira, André

doi:10.1016/j.crma.2005.01.005

Differential Geometry

Closed hypersurfaces of

S^{4} (1)

with constant mean curvature and zero Gauß–Kronecker curvature
[Hypersurfaces fermées de

S^{4} (1)

à courbure moyenne constante et à courbure de Gauß–Kronecker nulle]

Présenté par : Thierry Aubin

Lusala, Tsasa ¹ ; Gomes de Oliveira, André ¹

¹ Instituto de Matemática e Estatística (IME), Universidade de São Paulo (USP), Rua do Matão, 1010, Cidade Universitária, CEP 05508-090 São Paulo – SP, Brazil

Comptes Rendus. Mathématique, Tome 340 (2005) no. 6, pp. 437-440.

Résumé
Abstract

Nous considérons une hypersurface fermée (compacte et sans bord) $M^{3} \subset S^{4} (1)$ à courbure de Gauß–Kronecker identiquement nulle. Nous prouvons que si la courbure moyenne H de $M^{3}$ est constante, alors l'hypersurface $M^{3}$ est necéssairement minimale, c.à.d, $H = 0$ . Ce résultat généralise celui obtenu dans l'article de Ramanathan (Math. Z. 205 (1990) 645–658) concernant les hypersurfaces fermées minimales à courbure de Gauß–Kronecker identiquement nulle dans $S^{4} (1)$ .

We consider a closed hypersurface $M^{3} \subset S^{4} (1)$ with identically zero Gauß–Kronecker curvature. We prove that if $M^{3}$ has constant mean curvature H, then $M^{3}$ is minimal, i.e., $H = 0$ . This result extends Ramanathan's classification (Math. Z. 205 (1990) 645–658) result of closed minimal hypersurfaces of $S^{4} (1)$ with vanishing Gauß–Kronecker curvature.

Reçu le : 2004-09-19
Accepté le : 2004-12-12
Publié le : 2005-03-05

DOI : 10.1016/j.crma.2005.01.005

Affiliations des auteurs :

Lusala, Tsasa ¹ ; Gomes de Oliveira, André ¹

¹ Instituto de Matemática e Estatística (IME), Universidade de São Paulo (USP), Rua do Matão, 1010, Cidade Universitária, CEP 05508-090 São Paulo – SP, Brazil

@article{CRMATH_2005__340_6_437_0,
     author = {Lusala, Tsasa and Gomes de Oliveira, Andr\'e},
     title = {Closed hypersurfaces of $ {\mathbb{S}}^{4}(1)$ with constant mean curvature and zero {Gau{\ss}{\textendash}Kronecker} curvature},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {437--440},
     publisher = {Elsevier},
     volume = {340},
     number = {6},
     year = {2005},
     doi = {10.1016/j.crma.2005.01.005},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2005.01.005/}
}

TY  - JOUR
AU  - Lusala, Tsasa
AU  - Gomes de Oliveira, André
TI  - Closed hypersurfaces of $ {\mathbb{S}}^{4}(1)$ with constant mean curvature and zero Gauß–Kronecker curvature
JO  - Comptes Rendus. Mathématique
PY  - 2005
SP  - 437
EP  - 440
VL  - 340
IS  - 6
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2005.01.005/
DO  - 10.1016/j.crma.2005.01.005
LA  - en
ID  - CRMATH_2005__340_6_437_0
ER  -

%0 Journal Article
%A Lusala, Tsasa
%A Gomes de Oliveira, André
%T Closed hypersurfaces of $ {\mathbb{S}}^{4}(1)$ with constant mean curvature and zero Gauß–Kronecker curvature
%J Comptes Rendus. Mathématique
%D 2005
%P 437-440
%V 340
%N 6
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2005.01.005/
%R 10.1016/j.crma.2005.01.005
%G en
%F CRMATH_2005__340_6_437_0

Lusala, Tsasa; Gomes de Oliveira, André. Closed hypersurfaces of $ {\mathbb{S}}^{4}(1)$ with constant mean curvature and zero Gauß–Kronecker curvature. Comptes Rendus. Mathématique, Tome 340 (2005) no. 6, pp. 437-440. doi : 10.1016/j.crma.2005.01.005. http://www.numdam.org/articles/10.1016/j.crma.2005.01.005/

Bibliographie
Cité par

[1] de Almeida, S.; Brito, F. Minimal hypersurfaces of $S^{4}$ with constant Gauß–Kronecker curvature, Math. Z., Volume 195 (1987), pp. 99-107

[2] de Almeida, S.; Brito, F. Closed hypersurfaces with two constant curvatures, Ann. Fac. Sc. Toulouse Math. (4), Volume 6 (1997), pp. 187-202

[3] S. de Almeida, F. Brito, A. de Sousa, Hypersurfaces of $S^{4}$ with non-zero constant Gauss–Kronecker curvature, Preprint

[4] Brito, F.; Liu, H.L.; Simon, U.; Wang, C.P. Hypersurfaces in a space forms with some constant curvature functions (Defever, F. et al., eds.), Geometry and Topology of Submanifolds, vol. IX, World Scientific, Singapore, 1999, pp. 48-63

[5] Chang, S. A closed hypersurface with constant scalar curvature and constant mean curvature in $S^{4}$ is isoparametric, Commun. Anal. Geom., Volume 1 (1993), pp. 71-100

[6] Chang, S. On closed hypersurfaces of constant scalar curvatures and mean curvatures in $S^{n + 1}$ , Pacific J. Math., Volume 165 (1994), pp. 67-76

[7] Cheng, Q.-M. Complete hypersurfaces in a 4-dimensional unit sphere with constant mean curvature, Yokohama Math. J., Volume 29 (1992), pp. 125-139

[8] Peng, C.K.; Terng, C.L. Minimal hypersurfaces of spheres with constant scalar curvature (Bombieri, E., ed.), Seminar on Minimal Submanifolds, Ann. of Math. Stud., vol. 103, Princeton University Press, Princeton, NY, 1983, pp. 179-198

[9] Peng, C.K.; Terng, C.L. The scalar curvature of minimal hypersurfaces in spheres, Ann. Math., Volume 266 (1983), pp. 62-105

[10] Ramanathan, J. Minimal hypersurfaces in $S^{4}$ with vanishing Gauss–Kronecker curvature, Math. Z., Volume 205 (1990), pp. 645-658

Cité par Sources :