Complete minimal hypersurfaces in  with zero Gauss–Kronecker curvature

Hasanis, Thomas; Savas-Halilaj, Andreas; Vlachos, Theodoros

doi:10.1017/s0305004106009686

Cited by 8 publications

(14 citation statements)

References 10 publications

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“…Combining Theorems 1.2 and 1.4 with their results we obtain the following statements. We point out that Theorem 1.5(i) and (ii) are proved in [6,7], respectively, and (ii) of Theorem 1.6 is proved in [8].…”

Section: Theorem 14 Let F : M 3 → S 4 Be a Minimal Immersion Of A Comentioning

confidence: 92%

“…Hasanis et al [6][7][8], gave examples and a classification of complete minimal hypersurfaces in Q 4 (c) with K identically zero. Combining Theorems 1.2 and 1.4 with their results we obtain the following statements.…”

Section: Theorem 14 Let F : M 3 → S 4 Be a Minimal Immersion Of A Comentioning

confidence: 99%

“…Recently there has been a great interest in studying the complete minimal hypersurfaces in a space form Q 4 (c) with constant Gauss-Kronecker curvature, see the very interesting papers of Cheng [2] and Hasanis et al [6][7][8]. In particular, Cheng proved, in [2], that a complete minimal hypersurface in Q 4 (c), c ≤ 0, with scalar curvature R bounded from below and constant Gauss-Kronecker curvature K , must have K identically zero.…”

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confidence: 99%

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The Gauss–Kronecker curvature of minimal hypersurfaces in four-dimensional space forms

2009

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Let Q 4 (c) be a four-dimensional space form of constant curvature c. In this paper we show that the infimum of the absolute value of the Gauss-Kronecker curvature of a complete minimal hypersurface in Q 4 (c), c ≤ 0, whose Ricci curvature is bounded from below, is equal to zero. Further, we study the connected minimal hypersurfaces M 3 of a space form Q 4 (c) with constant Gauss-Kronecker curvature K . For the case c ≤ 0, we prove, by a local argument, that if K is constant, then K must be equal to zero. We also present a classification of complete minimal hypersurfaces of Q 4 (c) with K constant.

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Section: Theorem 14 Let F : M 3 → S 4 Be a Minimal Immersion Of A Comentioning

confidence: 92%

Section: Theorem 14 Let F : M 3 → S 4 Be a Minimal Immersion Of A Comentioning

confidence: 99%

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confidence: 99%

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The Gauss–Kronecker curvature of minimal hypersurfaces in four-dimensional space forms

2009

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“…Ramanathan [12] extended this result and allowed points where the second fundamental form is zero. In [7], [8] the authors extended the above results to complete minimal hypersurfaces in the Euclidean space R 4 or in the unit sphere S 4 . The aim of this paper is to study complete minimal hypersurfaces in the 4-dimensional hyperbolic space H 4 with identically zero Gauss-Kronecker curvature.…”

Section: Introductionmentioning

confidence: 88%

Complete minimal hypersurfaces in the hyperbolic space ℍ⁴ with vanishing Gauss-Kronecker curvature

Hasanis

Savas-Halilaj

Vlachos

2007

Trans. Amer. Math. Soc.

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“…From Cheng's result we see that the Gauss-Kronecker curvature must be zero. In [8], Hasanis, Savas-Halilaj and Vlachos gave a classification of such kind of hypersurfaces with assumption that the second fundamental form is nowhere vanishing. In fact, they obtained the following result.…”

Section: §1 Introductionmentioning

confidence: 99%

Complete hypersurfaces in a 4-dimensional hyperbolic space

Zhao

2009

Appl. Math. J. Chin. Univ.

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This paper gives a classification of complete hypersurfaces with nonzero constant mean curvature and constant quasi-Gauss-Kronecker curvature in the hyperbolic space H 4 (−1), whose scalar curvature is bounded from below. §1 Introduction Let M 3 be a 3-dimensional hypersurface in a simply connected space form F 4 (c) of constant sectional curvature c. Peng and Terng [1] and Cheng [2] proved that Cartan's isoparametric minimal hypersurfaces with zero Gauss-Kronecker curvature are the only complete minimal hypersurfaces in unit sphere S 4 (1) with constant scalar curvature and three distinct principal curvatures at each point. On the other hand, Almeida and Brito [3] considered the closed minimal hypersurfaces in S 4 (1) with constant Gauss-Kronecker curvature. They proved that the Clifford torus S 2 × S 1 and the boundary of the tube of a minimal 2-dimensional surface in S 4 (1) with geodesic radius π 2 are the only closed minimal hypersurfaces in S 4 (1) with constant GaussKronecker curvature and non-vanishing second fundamental form. Ramanathan [4] extended this result and allowed points where the second fundamental form is zero. Recently, Almeida, Brito and Sousa [5] completely classified the closed hypersurfaces of S 4 (1) where any two of the mean curvature, the scalar curvature and the Gauss-Kronecker curvature are constant. In [6], Hasanis, Savas-Halilaj and Vlachos extended the above results to complete minimal hypersurfaces in Euclidean space. It is natural to investigate complete hypersurfaces in hyperbolic space H n+1 (−1) of constant curvature −1 under some curvature conditions. Cheng [7] showed that the Gauss-Kronecker curvature of complete minimal hypersurfaces in H 4 (−1) with constant Gauss-Kronecker curvature must be zero if its scalar curvature is bounded from below. We know that the quasi-Gauss-Kronecker curvature is exactly the Gauss-Kronecker curvature Received: 2008-12-02 MR Subject Classification: 53C24, 53C40

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Complete minimal hypersurfaces in with zero Gauss–Kronecker curvature

Cited by 8 publications

References 10 publications

The Gauss–Kronecker curvature of minimal hypersurfaces in four-dimensional space forms

The Gauss–Kronecker curvature of minimal hypersurfaces in four-dimensional space forms

Complete minimal hypersurfaces in the hyperbolic space ℍ⁴ with vanishing Gauss-Kronecker curvature

Complete hypersurfaces in a 4-dimensional hyperbolic space

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