Hypersurfaces with constant mean curvature in spheres

This paper studies topological and metric rigidity theorems for hypersurfaces in a Euclidean sphere. We first show that an n( 2)-dimensional complete connected oriented closed hypersurface with non-vanishing Gauss-Kronecker curvature immersed in a Euclidean open hemisphere is diffeomorphic to a Euclidean n-sphere. We also show that an n( 2)-dimensional complete connected orientable hypersurface immersed in a unit sphere S n+1 whose Gauss image is contained in a closed geodesic ball of radius less than π/2 in S n+1 is diffeomorphic to a sphere. Finally, we prove that an n( 2)-dimensional connected closed orientable hypersurface in S n+1 with constant scalar curvature greater than n(n − 1) and Gauss image contained in an open hemisphere is totally umbilic.

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“…The authors were informed recently that Theorem 1.3 in this paper has also been proved, by a different method, in [2].…”

Section: Since (Nh)mentioning

confidence: 91%

Rigidity of Hypersurfaces in a Euclidean Sphere

Wang

Xia

2006

Proceedings of the Edinburgh Mathematical Society

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“…The first one is a classic algebraic lemma due to M. Okumura in [13], and completed with the equality case in [1] by H. Alencar and M. do Carmo. where β ≥ 0.…”

Section: Key Lemmasmentioning

confidence: 99%

On complete linear Weingarten hypersurfaces in locally symmetric Riemannian manifolds

Aquino¹,

Lima²,

Santos³

et al. 2015

Commentationes Mathematicae Universitatis Carolinae

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“…We must note that when k = 1 the operator φ 1 (X) = φ(X) = HX − AX was used in [1], where φ 1 ≡ 0 if and only if the immersion is totally umbilical. This fact extends to k-umbilical immersions: by (5.2) φ k ≡ 0 if and only if the immersion is k-umbilical; in another words, the operator φ k gives a measure of how much an isometric immersion fails to be k-umbilical.…”

Section: Proof For Any X Y Tangents Tomentioning

confidence: 99%

“…Remark 5.7). For k = 1 this was studied in [1]. An example of a two-umbilical embedding is given by S k (…”

Section: Introduction Let X : Mmentioning

confidence: 99%

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Constant Scalar Curvature Hypersurfaces With Second-Order Umbilicity

Colares

Echaiz-Espinoza

2009

Glasgow Math. J.

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Abstract.We extend the concept of umbilicity to higher order umbilicity in Riemannian manifolds saying that an isometric immersion is k-umbilical when AP k−1 (A) is a multiple of the identity, where P k (A) is the kth Newton polynomial in the second fundamental form A with P 0 (A) being the identity. Thus, for k = 1, oneumbilical coincides with umbilical. We determine the principal curvatures of the twoumbilical isometric immersions in terms of the mean curvatures. We give a description of the two-umbilical isometric immersions in space forms which includes the product of spheres) embedded in the Euclidean sphere S 2k+1 of radius 1. We also introduce an operator φ k which measures how an isometric immersion fails to be k-umbilical, giving in particular that φ 1 ≡ 0 if and only if the immersion is totally umbilical. We characterize the two-umbilical hypersurfaces of a space form as images of isometric immersions of Einstein manifolds.

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Hypersurfaces with constant mean curvature in spheres

Cited by 115 publications

References 7 publications

Rigidity of Hypersurfaces in a Euclidean Sphere

Rigidity of Hypersurfaces in a Euclidean Sphere

On complete linear Weingarten hypersurfaces in locally symmetric Riemannian manifolds

Constant Scalar Curvature Hypersurfaces With Second-Order Umbilicity

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