A Note on Minimal Hypersurfaces of an Odd Dimensional Sphere

Deshmukh, Sharief; Al-Dayel, İbrahim

doi:10.3390/math8020294

Cited by 2 publications

(1 citation statement)

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“…Minimal hypersurfaces in a unit sphere is a very important subject in differential geometry that has been investigated by many researchers (cf. [1][2][3][4][5][6][7][8][9][10][11]). An important property of these hypersurfaces is that, if the shape operator A of a minimal compact hypersurface M of S n+1 satisfies A 2 < n, then it is totally geodesic and if A 2 = n, then it is a Clifford hypersurface (cf.…”

Section: Introductionmentioning

confidence: 99%

On Minimal Hypersurfaces of a Unit Sphere

et al. 2021

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Minimal compact hypersurface in the unit sphere Sn+1 having squared length of shape operator A2<n are totally geodesic and with A2=n are Clifford hypersurfaces. Therefore, classifying totally geodesic hypersurfaces and Clifford hypersurfaces has importance in geometry of compact minimal hypersurfaces in Sn+1. One finds a naturally induced vector field w called the associated vector field and a smooth function ρ called support function on the hypersurface M of Sn+1. It is shown that a necessary and sufficient condition for a minimal compact hypersurface M in S5 to be totally geodesic is that the support function ρ is a non-trivial solution of static perfect fluid equation. Additionally, this result holds for minimal compact hypersurfaces in Sn+1, (n>2), provided the scalar curvature τ is a constant on integral curves of w. Yet other classification of totally geodesic hypersurfaces among minimal compact hypersurfaces in Sn+1 is obtained using the associated vector field w an eigenvector of rough Laplace operator. Finally, a characterization of Clifford hypersurfaces is found using an upper bound on the integral of Ricci curvature in the direction of the vector field Aw.

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Section: Introductionmentioning

confidence: 99%

On Minimal Hypersurfaces of a Unit Sphere

et al. 2021

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Investigating the characteristics of Clifford hypersurfaces and the unit sphere via a minimal immersion in $ S^{n+1} $

Al-dayel

2024

MATH

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<p>In this article, we find the different sufficient conditions for a compact minimal hypersurface $ M $ of the unit sphere $ S^{n+1}, n\in \mathbb{Z}^{+} $ to be the Clifford hypersurface $ S^{\ell }(\sqrt{\frac{\ell }{n}})\times S^{m}(\sqrt{\frac{m}{n}}), $ where $ \ell, m\in \mathbb{Z}^{+}, \; \ell +m = n $ or the sphere $ S^{n} $. This classification is achieved by applying constraints to the tangent and normal components of the immersion.</p>

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A Note on Minimal Hypersurfaces of an Odd Dimensional Sphere

Abstract: We obtain the Wang-type integral inequalities for compact minimal hypersurfaces in the unit sphere S 2 n + 1 with Sasakian structure and use these inequalities to find two characterizations of minimal Clifford hypersurfaces in the unit sphere S 2 n + 1 .

Cited by 2 publications

References 13 publications

On Minimal Hypersurfaces of a Unit Sphere

On Minimal Hypersurfaces of a Unit Sphere

Investigating the characteristics of Clifford hypersurfaces and the unit sphere via a minimal immersion in $ S^{n+1} $

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