Constant mean curvature Isometric Immersions into $\protect \mathbb{S}^2 \times \protect \mathbb{R}$ and $\protect \mathbb{H}^2 \times \protect \mathbb{R}$ and related results

Daniel, Benoît; Domingos, Iury; Vitório, Feliciano

doi:10.5802/aif.3521

Constant mean curvature Isometric Immersions into

𝕊^{2} \times ℝ

and

ℍ^{2} \times ℝ

and related results
[Immersions isométriques à courbure moyenne constante dans

𝕊^{2} \times ℝ

ℍ^{2} \times ℝ

et résultats apparentés]

Daniel, Benoît ¹ ; Domingos, Iury ^1,² ; Vitório, Feliciano ²

¹ Université de Lorraine CNRS, IECL 54000 Nancy (France)
² Universidade Federal de Alagoas Instituto de Matemática Campus A. C. Simões BR 104 - Norte Km 97 57072-970, Maceió - AL (Brazil)

Annales de l'Institut Fourier, Tome 73 (2023) no. 1, pp. 203-249.

Résumé
Abstract

Dans cet article, nous étudions les immersions à courbure moyenne constante dans $𝕊^{2} \times ℝ$ et $ℍ^{2} \times ℝ$ et nous classifions ces immersions quand la surface est à courbure intrinsèque constante. Comme applications, nous utilisons la correspondance des surfaces sœurs pour classifier les surfaces à courbure moyenne constante et courbure intrinsèque constante dans les variétés de dimension $3$ homogènes $𝔼 (κ, τ)$ et nous utilisons la correspondance de Torralbo–Urbano pour classifier les surfaces à courbure moyenne parallèle et courbure intrinsèque constante dans $𝕊^{2} \times 𝕊^{2}$ et $ℍ^{2} \times ℍ^{2}$ . Il est important de noter que ces classifications fournissent de nouveaux exemples.

In this article, we study constant mean curvature isometric immersions into $𝕊^{2} \times ℝ$ and $ℍ^{2} \times ℝ$ and we classify these isometric immersions when the surface has constant intrinsic curvature. As applications, we use the sister surface correspondence to classify the constant mean curvature surfaces with constant intrinsic curvature in the $3$ -dimensional homogenous manifolds $𝔼 (κ, τ)$ and we use the Torralbo–Urbano correspondence to classify the parallel mean curvature surfaces in $𝕊^{2} \times 𝕊^{2}$ and $ℍ^{2} \times ℍ^{2}$ with constant intrinsic curvature. It is worthwhile to point out that these classifications provide new examples.

Reçu le : 2019-12-13
Révisé le : 2021-05-19
Accepté le : 2021-06-23
Publié le : 2023-05-12

DOI : 10.5802/aif.3521

Classification : 53C42, 53A10, 53C30
Keywords: Isometric immersions, constant mean curvature surfaces, parallel mean curvature surfaces, homogenous $3$-manifolds.
Mot clés : Immersions isométriques, surfaces à courbure moyenne constante, surfaces à courbure moyenne parallèle, variétés homogènes de dimension $3$.

Affiliations des auteurs :

Daniel, Benoît ¹ ; Domingos, Iury ^{1, 2} ; Vitório, Feliciano ²

¹ Université de Lorraine CNRS, IECL 54000 Nancy (France)
² Universidade Federal de Alagoas Instituto de Matemática Campus A. C. Simões BR 104 - Norte Km 97 57072-970, Maceió - AL (Brazil)

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Constant mean curvature {Isometric} {Immersions} into $\protect \mathbb{S}^2 \times \protect \mathbb{R}$ and $\protect \mathbb{H}^2 \times \protect \mathbb{R}$ and related results},
     journal = {Annales de l'Institut Fourier},
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Daniel, Benoît; Domingos, Iury; Vitório, Feliciano. Constant mean curvature Isometric Immersions into $\protect \mathbb{S}^2 \times \protect \mathbb{R}$ and $\protect \mathbb{H}^2 \times \protect \mathbb{R}$ and related results. Annales de l'Institut Fourier, Tome 73 (2023) no. 1, pp. 203-249. doi : 10.5802/aif.3521. https://aif.centre-mersenne.org/articles/10.5802/aif.3521/

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