Minimal Submanifolds of a Sphere with Second Fundamental Form of Constant Length

Chern, Shiing-Shen; Carmo, Manfredo do; Kobayashi, Shimpei

doi:10.1007/978-3-642-25588-5_5

Cited by 240 publications

(132 citation statements)

References 2 publications

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“…. , n + 2 and so, using lemma 1 we conclude that (n−|σ| 2 ) f, a i = 0 for any i, which is possible only if n−|σ| 2 = 0 on M. Now the result of Chern, do Carmo and Kobayashi [3] says that M is locally congruent to a Clifford minimal hypersurface. Thus M is congruent either to the Clifford hypersurface…”

mentioning

confidence: 84%

“…Previously do Carmo and Peng [4] and Fischer-Colbrie and Schoen [6] had shown that the only stable (index zero) complete minimal surface is the plane. From the work by Ritoré and Ros [11] a classification of index one minimal surfaces in RP 3 (1) can be obtained: its must be a two-fold covering of a linear subvariety or a tube of certain radius around a line. These authors ( [12]) also obtained a compactness for the space of index one minimal surfaces in flat three tori.…”

Section: Introductionmentioning

confidence: 99%

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Compact minimal hypersurfaces with index one in the real projective space

Carmo

Ritoré

Ros

2012

Manfredo P. Do Carmo – Selected Papers

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Let M n be a compact (two-sided) minimal hypersurface in a Riemannian manifold M n+1. It is a simple fact that if M has positive Ricci curvature then M cannot be stable (i. e. its Jacobi operator L has index at least one). If M = S n+1 (1) is the unit sphere and L has index one, then it is known that M must be a totally geodesic equator. We prove that if M is the real projective space RP n+1 = S n+1 (1)/{±}, obtained as a metric quotient of the unit sphere, and the Jacobi operator of M has index one, then M is either a totally geodesic sphere or the quotient to the projective space of the hypersurface S n1 (R 1) × S n2 (R 2) ⊂ S n+1 (1) obtained as the product of two spheres of dimensions n

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mentioning

confidence: 84%

Section: Introductionmentioning

confidence: 99%

Compact minimal hypersurfaces with index one in the real projective space

Carmo

Ritoré

Ros

2012

Manfredo P. Do Carmo – Selected Papers

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“…Theorem 3 in [4]). Note that this result was independently proven by Lawson [2] and Chern, do Carmo, and Kobayashi [5]. One of the interesting questions in differential geometry of minimal hypersurfaces of the unit sphere S n+1 is to characterize minimal Clifford hypersurfaces.…”

Section: Introductionmentioning

confidence: 74%

A Note on Minimal Hypersurfaces of an Odd Dimensional Sphere

Deshmukh

Al-Dayel

2020

Mathematics

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“…Example 3.2. From [5] and [1], we know that the Veronese surface is a minimal surface in 4 ( ) which is embeded in 4+ ( ) as a totally geodesic spacelike submanifold such that 2 + = , then the Veronese surface is a maximal spacelike surface in 2+ ( ), where = 2 + . From Proposition 3.1, we know that it is a Willmore spacelike surface in 4+ ( ).…”

Section: Euler-lagrange Equation Of Willmore Spacelike Submanifoldsmentioning

confidence: 99%

Willmore spacelike submanifolds in an indefinite space form Nn+p q(c)

Шу

Chen

2017

Publ Inst Math (Belgr)

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Let Nn+p q(c) be an (n+p)-dimensional connected indefinite space form of index q(1 ? q ? p) and of constant curvature c. Denote by ? : M ? Nn+p q (c) the n-dimensional spacelike submanifold in Nn+p q (c), ? : M ? Nn+p q(c) is called a Willmore spacelike submanifold in Nn+p q(c) if it is a critical submanifold to the Willmore functional W(?) = ?q M ?n dv =?M (S-nH2)n/2 dv, where S and H denote the norm square of the second fundamental form and the mean curvature of M and ?2 = S ? nH2. If q = p, in [14], we proved some integral inequalities of Simons? type and rigidity theorems for n-dimensional Willmore spacelike submanifolds in a Lorentzian space form Nn+p q(c). In this paper, we continue to study this topic and prove some integral inequalities of Simons? type and rigidity theorems for n-dimensional Willmore spacelike submanifolds in an indefinite space form Nn+p q(c) (1 ? q ? p).

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Minimal Submanifolds of a Sphere with Second Fundamental Form of Constant Length

Cited by 240 publications

References 2 publications

Compact minimal hypersurfaces with index one in the real projective space

Compact minimal hypersurfaces with index one in the real projective space

A Note on Minimal Hypersurfaces of an Odd Dimensional Sphere

Willmore spacelike submanifolds in an indefinite space form Nn+p q(c)

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