Hypersurfaces of constant higher-order mean curvature in $$M\times {\mathbb{R}}$$

Lima, Ronaldo F. de; Manfio, Fernando; Santos, João Paulo dos

doi:10.1007/s10231-022-01229-3

Cited by 7 publications

(8 citation statements)

References 24 publications

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“…This fact, together with the main results in [7] (see also [15,23]), allows us to apply the Alexandrov reflection method to provide uniqueness results for the rotational elliptic Weingarten spheres of ℚ 𝑛 𝜖 × ℝ (𝜖 ≠ 0, 𝑛 ≥ 3) we constructed in Section 5. Similar results were obtained in [8] for hypersurfaces of constant higher order mean curvatures. Definition 7.1.…”

Section: Uniqueness Of Rotational Elliptic Weingarten Spheressupporting

confidence: 86%

“…Let us see now that the compactness hypothesis in Theorem 7.2 can be replaced by completeness if we add conditions on the height function 𝜉 of 𝛴 and on its second fundamental form. This is accomplished by means of the following general height estimate, obtained in [8]. Lemma 7.3 [8,Proposition 3].…”

Section: Uniqueness Of Rotational Elliptic Weingarten Spheresmentioning

confidence: 99%

“…This is accomplished by means of the following general height estimate, obtained in [8]. Lemma 7.3 [8,Proposition 3]. Consider an arbitrary Riemannian manifold 𝑀, and let 𝛴 ⊂ 𝑀 × ℝ be a compact vertical graph of a nonnegative function defined on a domain Ω ⊂ 𝑀 × {0}.…”

Section: Uniqueness Of Rotational Elliptic Weingarten Spheresmentioning

confidence: 99%

“…Let 𝑀 be either ℍ 𝑚 𝔽 of 𝕊 𝑛 . It was proved in [8] that, for all 𝑐 > 0, the elliptic Weingarten function 𝑊 𝑐 = 𝐻 𝑟 − 𝑐 is 𝑀admissible. Moreover, for 𝑀 = ℍ 𝑚 𝔽 , there is a constant 𝐶(𝔽) > 0 such that the associated function 𝜚 𝑐 to 𝑊 𝑐 satisfies the conditions of Theorem 5.3-(i) (resp.…”

mentioning

confidence: 99%

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Elliptic Weingarten hypersurfaces of Riemannian products

Lima

Ramos

Santos³

2023

Mathematische Nachrichten

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Let 𝑀 𝑛 be either a simply connected space form or a rank-one symmetric space of the noncompact type. We consider Weingarten hypersurfaces of 𝑀 × ℝ, which are those whose principal curvatures 𝑘 1 , … , 𝑘 𝑛 and angle function 𝛩 satisfy a relation 𝑊(𝑘 1 , … , 𝑘 𝑛 , 𝛩 2 ) = 0, being 𝑊 a differentiable function which is symmetric with respect to 𝑘 1 , … , 𝑘 𝑛 . When 𝜕𝑊∕𝜕𝑘 𝑖 > 0 on the positive cone of ℝ 𝑛 , a strictly convex Weingarten hypersurface determined by 𝑊 is said to be elliptic. We show that, for a certain class of Weingarten functions 𝑊, there exist rotational strictly convex Weingarten hypersurfaces of 𝑀 × ℝ which are either topological spheres or entire graphs over 𝑀. We establish a Jellett-Liebmanntype theorem by showing that a compact, connected and elliptic Weingarten hypersurface of either 𝕊 𝑛 × ℝ or ℍ 𝑛 × ℝ is a rotational embedded sphere. Other uniqueness results for complete elliptic Weingarten hypersurfaces of these ambient spaces are obtained. We also obtain existence results for constant scalar curvature hypersurfaces of 𝕊 𝑛 × ℝ and ℍ 𝑛 × ℝ which are either rotational or invariant by translations (parabolic or hyperbolic). We apply our methods to give new proofs of the main results by Manfio and Tojeiro on the classification of constant sectional curvature hypersurfaces of 𝕊 𝑛 × ℝ and ℍ 𝑛 × ℝ.

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Section: Uniqueness Of Rotational Elliptic Weingarten Spheressupporting

confidence: 86%

Section: Uniqueness Of Rotational Elliptic Weingarten Spheresmentioning

confidence: 99%

Section: Uniqueness Of Rotational Elliptic Weingarten Spheresmentioning

confidence: 99%

mentioning

confidence: 99%

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Elliptic Weingarten hypersurfaces of Riemannian products

Lima

Ramos

Santos³

2023

Mathematische Nachrichten

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“…In this section, we shall approach the rotational CMC spheres Σ H of Q n × R as in [4], where the more general case of hypersurfaces of constant higher order mean curvature was considered (see also [5]).…”

Section: Proofs Of Theorem 1 and Corollaries 1 Andmentioning

confidence: 99%

On Stable Constant Mean Curvature Spheres of $\mathbb H^n\times\mathbb R$ and $\mathbb S^n\times\mathbb R$ and their Uniqueness as Isoperimetric Hypersurfaces

Lima¹,

Elbert²,

Nelli³

2023

Preprint

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We consider the rotational constant mean curvature spheres of H n × R obtained by Hsiang and Hsiang and show that they are nested. We apply this property to prove the uniqueness of these spheres as isoperimetric hypersurfaces of H n × R, filling in a gap in the original proof given by Hsiang and Hsiang. We also give a direct proof of their stability, extending the result by Souam, who considered the case n = 2. Analogous results are obtained for the class of the rotational constant large mean curvature spheres of S n × R constructed by Pedrosa.

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