A characterization of Clifford hypersurfaces among embedded constant mean curvature hypersurfaces in a unit sphere

Min, Sung-Hong; Seo, Keomkyo

doi:10.4310/mrl.2017.v24.n2.a12

Cited by 6 publications

(3 citation statements)

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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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“…In his pioneering work, Simons [1] has shown that on a compact minimal hypersurface M of the unit sphere S n+1 either A = 0 (totally geodesic), or A 2 = n, or A 2 (p) > n for some point p ∈ M, where A is the length of the shape operator. This work was further extended in [2] and for compact constant mean curvature hypersurfaces in [3]. If for every point p in M, the square of the length of the second fundamental form of M is n, then it is known that M must be a subset of a Clifford minimal hypersurface…”

Section: Introductionmentioning

confidence: 99%

“…In [2], bounds on Ricci curvature are used to find a characterization of the minimal Clifford hypersurfaces in the unit sphere S 4 . Similarly in [3,[9][10][11], the authors have characterized minimal Clifford hypersurfaces in the odd-dimensional unit spheres S 3 and S 5 using constant contact angle. Wang [12] studied compact minimal hypersurfaces in the unit sphere S n+1 with two distinct principal curvatures, one of them being simple and obtained the following integral inequality,…”

Section: Introductionmentioning

confidence: 99%