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

Geodesic

Parcourir par

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.

Voir la notice de l'article provenant de la source Numdam

Abstract (VO)
Résumé

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.

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 $S^{4} (1)$ .

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

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     title = {Closed hypersurfaces of $ {\mathbb{S}}^{4}(1)$ with constant mean curvature and zero {Gau{\ss}{\textendash}Kronecker} curvature},
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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://geodesic.mathdoc.fr/articles/10.1016/j.crma.2005.01.005/

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