On the umbilicity of generalized linear Weingarten spacelike hypersurfaces in a Lorentzian space form

Aquino, Cícero P.; Batista, Márcio; Lima, Henrique F. de

doi:10.1016/j.geomphys.2018.12.006

Cited by 4 publications

(3 citation statements)

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“…, n − 1} are just the hypersurfaces having constant (r + 1)-th mean curvature H r+1 . In recent years, several papers have been published showing the interest in understanding the geometry of the (r, s)-linear Weingarten hypersurfaces (see [2,3,14,15,23]). For instance, we can highlight that the author jointly with H. de Lima and A. de Sousa showed in [23, Section 3] that (r, s)-linear Weingarten closed hypersurfaces compact are critical points of the variational problem of minimizing a suitable linear combination…”

Section: And (Cf [22 Teorema 2])mentioning

confidence: 99%

“…Taking into account the relation between H 2 and the normalized scalar curvature R given in (2.4), we observe from (2.10) that the (0, 1)-linear Weingarten hypersurfaces x : Σ n ↬ S n+1 are called simply linear Weingarten hypersurfaces, and there is a vast recent literature treating the problem of characterizing these hypersurfaces (see, for instance, [5,6,7,12,16,17,18]). It is because of this observation that the hypersurfaces described in Definition 2.1 are also called, in the current literature, the generalized linear Weingarten hypersurfaces (see [2,3,14,15,23]). Furthermore, when r = s ∈ {0, .…”

Section: And (Cf [22 Teorema 2])mentioning

confidence: 99%

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The region of the unit Euclidean sphere that admits a class of (r, s)-linear Weingarten hypersurfaces

Lázaro Velásquez

2023

Sel.mat

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In the unit Euclidean sphere Sn+1, we deal with a class of hypersurfaces that were characterized in [23] as the critical points of a variational problem, the so-called (r, s)-linear Weingarten hypersurfaces (0 ≤ r ≤s ≤ n−1); namely, the hypersurfaces of Sn+1 that has a linear combination arHr+1+...+asHs+1 of their higher order mean curvatures Hr+1 and Hs+1 being a real constant, where ar,..., ar are nonnegative real numbers (with at least one non zero). By assuming a geometric constraint involving the higher order mean curvatures of these hypersurfaces, we prove a uniqueness result for strongly stable (r, s)-linear Weingarten hypersurfaces immersed in a certain region determined by a geodesic sphere of Sn+1. We also establish a nonexistence result in another region of Sn+1 for strongly stable Weingarten (r, s)-linear hypersurfaces.

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Section: And (Cf [22 Teorema 2])mentioning

confidence: 99%

Section: And (Cf [22 Teorema 2])mentioning

confidence: 99%

The region of the unit Euclidean sphere that admits a class of (r, s)-linear Weingarten hypersurfaces

Lázaro Velásquez

2023

Sel.mat

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show abstract

“…, n − 1} are just the hypersurfaces having constant (r+1)-th mean curvature H r+1 . In recent years, several papers have been published showing the interest in understanding the geometry of the (r, s)-linear Weingarten hypersurfaces (see [1,2,13,14,15,23]). For instance, we can highlight that the author jointly with H. de Lima and A. de Sousa showed in [23, Section 3] that (r, s)-linear Weingarten compact hypersurfaces compact are critical points of the variational problem of minimizing a suitable linear combination a r A r + • • • + a s A s of the j-area functionals A j of a given compact oriented hypersurface Σ n ↬ S n+1 , j ∈ {r, .…”

mentioning

confidence: 99%

On the index of stability of (r, s)-linear Weingarten Clifford tori

Velásquez

2023

Sel.mat

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For entire numbers r and s satisfying 0 ≤ r ≤ s ≤ n − 2, we showed that the index of (r, s)-stability of a (r, s)-linear Weingarten Clifford torus immersed into the (n + 1)-dimensional unit Euclidean sphere, that has a linear combination of their higher order mean curvatures Hr+1 and Hs+1 being null, is exactly equal to n + 3 provided that a geometric condition involving Hr+2 and Hs+2 is satisfied.

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Half-space results for generalized linear Weingarten hypersurfaces immersed in the real projective space

Velásquez,

de Lima,

de Lacerda

2024

Bol. Soc. Mat. Mex.

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On the umbilicity of generalized linear Weingarten spacelike hypersurfaces in a Lorentzian space form

Cited by 4 publications

References 30 publications

The region of the unit Euclidean sphere that admits a class of (r, s)-linear Weingarten hypersurfaces

The region of the unit Euclidean sphere that admits a class of (r, s)-linear Weingarten hypersurfaces

On the index of stability of (r, s)-linear Weingarten Clifford tori

Half-space results for generalized linear Weingarten hypersurfaces immersed in the real projective space

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