2008
DOI: 10.1007/s12220-008-9029-8
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A Characterization of Quadric Constant Mean Curvature Hypersurfaces of Spheres

Abstract: Let φ : M → S n+1 ⊂ R n+2 be an immersion of a complete n-dimensional oriented manifold. For any v ∈ R n+2 , let us denote by v : M → R the function givenWe will prove that if M has constant mean curvature, and, for some v = 0 and some real number λ, we have that v = λf v , then, φ(M) is either a totally umbilical sphere or a Clifford hypersurface. As an application, we will use this result to prove that the weak stability index of any compact constant mean curvature hypersurface M n in S n+1 which is neither … Show more

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Cited by 15 publications
(19 citation statements)
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“…Using the same arguments as in [3] we can conclude that M is either a totally umbilical sphere or a Clifford hypersurface. Hence, I 3 (x) = ∅ for every x ∈ N .…”
Section: Proofs Of Theoremsmentioning
confidence: 69%
See 1 more Smart Citation
“…Using the same arguments as in [3] we can conclude that M is either a totally umbilical sphere or a Clifford hypersurface. Hence, I 3 (x) = ∅ for every x ∈ N .…”
Section: Proofs Of Theoremsmentioning
confidence: 69%
“…From [3], we know that principal curvatures of M along the integral curve of v T are Without loss of generality, we will assume that M is not totally umbilical.…”
Section: Proofs Of Theoremsmentioning
confidence: 99%
“…In [3], Alías, Brasil and Perdomo proved that the weak index of any other compact CMC hypersurface M in S m+1 which is not totally umbilical and has constant scalar curvature is greater than or equal to m + 2, with equality if and only if M is a CMC Clifford torus S j (r) × S m−j ( √ 1 − r 2 ) with radius j/(m + 2) ≤ r ≤ (j + 2)/(m + 2). More recently, in [4] the same authors complemented that estimate by showing that the weak index of any compact CMC hypersurface M in S m+1 which is neither totally umbilical nor a CMC Clifford torus and has constant scalar curvature is greater than or equal to 2m + 4. At this respect, it is worth pointing out that the weak stability index of the CMC Clifford torus S j (r) × S m−j ( √ 1 − r 2 ) depends on r reaching its minimum value m + 2 when r ∈ [ j/(m + 2), (j + 2)/(m + 2)], and converging to +∞ as r converges either to 0 or 1 (see [3] and Section 4 for the details).…”
Section: Introductionmentioning
confidence: 91%
“…In [1], Alías, Brasil and the first author proved that if M is a complete hypersurface with constant mean curvature and l v = λf v for some nonzero vector v ∈ R n+2 and some real number λ, then M is either totally umbilical (a Euclidean sphere) or M is a cartesian product of Euclidean spheres. Thit is, M must be a isoparametric hypersurface of order 1 or 2.…”
Section: Introductionmentioning
confidence: 99%