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

Tsasa Lusala; André Gomes de Oliveira

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

Presented by: Thierry Aubin

Tsasa Lusala ¹; André Gomes de Oliveira 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, Volume 340 (2005) no. 6, pp. 437-440

Abstract (Original language)
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 $S^{4} (1)$ .

Received: 2004-09-19
Accepted: 2004-12-12
Published online: 2005-03-05

DOI: 10.1016/j.crma.2005.01.005

Author's affiliations:

Tsasa Lusala ¹; André Gomes de Oliveira 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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     author = {Tsasa Lusala and Andr\'e Gomes de Oliveira},
     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},
     year = {2005},
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     language = {en},
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Tsasa Lusala; André Gomes de Oliveira. Closed hypersurfaces of $ {\mathbb{S}}^{4}(1)$ with constant mean curvature and zero Gauß–Kronecker curvature. Comptes Rendus. Mathématique, Volume 340 (2005) no. 6, pp. 437-440. doi: 10.1016/j.crma.2005.01.005

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