Rotational linear Weingarten surfaces into the Euclidean sphere

Barros, A.; Silva, Juscelino; Sousa, P.

doi:10.1007/s11856-012-0053-9

Cited by 14 publications

(9 citation statements)

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“…Thereafter, Shu [19] proved another rigidity theorems concerning linear Weingarten hypersurfaces with two distinct principal curvatures immersed in a space form ‫ޑ‬ n+1 c . We also point out that López [13], and Barros, Silva and Sousa [5,6] obtained descriptions related to rotational linear Weingarten surfaces in the Euclidean space and in the Euclidean sphere, respectively. Furthermore, the first and second authors [4] used the Hopf's strong maximum principle and an extension of a suitable maximum principle of Yau [21] due to Caminha [8] in order to study the geometry of complete linear Weingarten hypersurfaces with nonnegative sectional curvature immersed in the hyperbolic space ‫ވ‬ n+1 .…”

Section: Introduction and Statemeof The Main Resultsmentioning

confidence: 74%

Linear Weingarten Hypersurfaces With Bounded Mean Curvature in the Hyperbolic Space

2014

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We apply appropriate maximum principles in order to obtain characterization results concerning complete linear Weingarten hypersurfaces with bounded mean curvature in the hyperbolic space. By supposing a suitable restriction on the norm of the traceless part of the second fundamental form, we show that such a hypersurface must be either totally umbilical or isometric to a hyperbolic cylinder, when its scalar curvature is positive, or to a spherical cylinder, when its scalar curvature is negative. Related to the compact case, we also establish a rigidity result.2000 Mathematics Subject Classification. Primary 53C42; Secondary 53C40. Introduction and statemeof the main results.In the theory of isometric immersions, the study of complete hypersurfaces with constant scalar curvature immersed in a Riemannian space constitutes an important theme. In the seminal paper [10], Cheng and Yau introduced a new self-adjoint differential operator acting on smooth functions defined on Riemannian manifolds. As a by-product of such approach they were able to classify closed hypersurfaces with constant normalized scalar curvature R satisfying R ≥ c and nonnegative sectional curvature immersed in a real space form ‫ޑ‬ n+1

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Section: Introduction and Statemeof The Main Resultsmentioning

confidence: 74%

Linear Weingarten Hypersurfaces With Bounded Mean Curvature in the Hyperbolic Space

2014

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“…In a recent paper Li et al [6] show in Theorem 1.3 that under additional conditions on the sectional curvatures a standard sphere and the Clifford torus are the only Weingarten hypersurfaces of the Euclidean sphere S n+1 . For Weingarten surfaces in space forms we direct the reader to [4], [2], [1], [7] and [9]. Our purpose here is to present a complete description of rotational Weingarten hypersurfaces contained in the Euclidean sphere S n+1 under the additional condition…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Rotational linear Weingarten hypersurfaces in the Euclidean sphere Sⁿ⁺¹

Barros

Silva

Sousa

2014

Advances in Geometry

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“…We observe that, when r = 0, s = 1, and F = 1, these hypersurfaces are classically called linear Weingarten hypersurfaces and, in the last years, a vast literature has been produced in the direction to obtain characterization results of them (see, for instance, [2,3,7,8,20,26]). In this paper we extend this study to the anisotropic case.…”

Section: 3)mentioning

confidence: 99%

Stability of generalized linear Weingarten hypersurfaces immersed in the Euclidean space

Silva¹,

Lima²,

Velásquez³

2018

Publicacions Matemàtiques

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Given a positive function F defined on the unit Euclidean sphere and satisfying a suitable convexity condition, we consider, for hypersurfaces M n immersed in the Euclidean space R n+1 , the so-called k-th anisotropic mean curvatures H F k , 0 ≤ k ≤ n. For fixed 0 ≤ r ≤ s ≤ n, a hypersurface M n of R n+1 is said to be (r, s, F )-linear Weingarten when its k-th anisotropic mean curvatures H F k , r ≤ k ≤ s, are linearly related. In this setting, we establish the concept of stability concerning closed (r, s, F )-linear Weingarten hypersurfaces immersed in R n+1 and, afterwards, we prove that such a hypersurface is stable if, and only if, up to translations and homotheties, it is the Wulff shape of F . For r = s and F ≡ 1, our results amount to the standard stability studied, for instance, by Alencar-do Carmo-Rosenberg [1].

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Rotational linear Weingarten surfaces into the Euclidean sphere

Cited by 14 publications

References 10 publications

Linear Weingarten Hypersurfaces With Bounded Mean Curvature in the Hyperbolic Space

Linear Weingarten Hypersurfaces With Bounded Mean Curvature in the Hyperbolic Space

Rotational linear Weingarten hypersurfaces in the Euclidean sphere Sⁿ⁺¹

Stability of generalized linear Weingarten hypersurfaces immersed in the Euclidean space

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Rotational linear Weingarten surfaces into the Euclidean sphere

Cited by 14 publications

References 10 publications

Linear Weingarten Hypersurfaces With Bounded Mean Curvature in the Hyperbolic Space

Linear Weingarten Hypersurfaces With Bounded Mean Curvature in the Hyperbolic Space

Rotational linear Weingarten hypersurfaces in the Euclidean sphere Sn+1

Stability of generalized linear Weingarten hypersurfaces immersed in the Euclidean space

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Rotational linear Weingarten hypersurfaces in the Euclidean sphere Sⁿ⁺¹