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

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, Volume 73 (2023) no. 1, pp. 203-249.

Abstract
Résumé

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.

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.

Received: 2019-12-13
Revised: 2021-05-19
Accepted: 2021-06-23
Published online: 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 $3$.

Author's affiliations:

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)

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{AIF_2023__73_1_203_0,
     author = {Daniel, Beno{\^\i}t and Domingos, Iury and Vit\'orio, Feliciano},
     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},
     pages = {203--249},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {73},
     number = {1},
     year = {2023},
     doi = {10.5802/aif.3521},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3521/}
}

TY  - JOUR
AU  - Daniel, Benoît
AU  - Domingos, Iury
AU  - Vitório, Feliciano
TI  - 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
JO  - Annales de l'Institut Fourier
PY  - 2023
SP  - 203
EP  - 249
VL  - 73
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3521/
DO  - 10.5802/aif.3521
LA  - en
ID  - AIF_2023__73_1_203_0
ER  -

%0 Journal Article
%A Daniel, Benoît
%A Domingos, Iury
%A Vitório, Feliciano
%T 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
%J Annales de l'Institut Fourier
%D 2023
%P 203-249
%V 73
%N 1
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3521/
%R 10.5802/aif.3521
%G en
%F AIF_2023__73_1_203_0

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, Volume 73 (2023) no. 1, pp. 203-249. doi : 10.5802/aif.3521. https://aif.centre-mersenne.org/articles/10.5802/aif.3521/

References
Cited by

[1] Abresch, Uwe; Rosenberg, Harold A Hopf differential for constant mean curvature surfaces in $S^{2} \times R$ and $H^{2} \times R$ , Acta Math., Volume 193 (2004) no. 2, pp. 141-174 | DOI | MR | Zbl

[2] Abresch, Uwe; Rosenberg, Harold Generalized Hopf differentials, Mat. Contemp., Volume 28 (2005), pp. 1-28 | MR | Zbl

[3] Aledo, Juan A.; Espinar, José M.; Gálvez, José A. Complete surfaces of constant curvature in $H^{2} \times ℝ$ and $S^{2} \times ℝ$ , Calc. Var. Partial Differential Equations, Volume 29 (2007) no. 3, pp. 347-363 | DOI | MR | Zbl

[4] Aledo, Juan A.; Espinar, José M.; Gálvez, José A. Surfaces with constant curvature in $𝕊^{2} \times ℝ$ and $ℍ^{2} \times ℝ$ . Height estimates and representation, Bull. Braz. Math. Soc. (N.S.), Volume 38 (2007) no. 4, pp. 533-554 | DOI | MR | Zbl

[5] Cartan, Élie Familles de surfaces isoparamétriques dans les espaces à courbure constante, Ann. Mat. Pura Appl., Volume 17 (1938) no. 1, pp. 177-191 | DOI | MR | Zbl

[6] Chen, Bang-yen Minimal surfaces with constant Gauss curvature, Proc. Amer. Math. Soc., Volume 34 (1972), pp. 504-508 | DOI | MR | Zbl

[7] Daniel, Benoît Isometric immersions into 3-dimensional homogeneous manifolds, Comment. Math. Helv., Volume 82 (2007) no. 1, pp. 87-131 | DOI | MR | Zbl

[8] Daniel, Benoît Isometric immersions into $𝕊^{n} \times ℝ$ and $ℍ^{n} \times ℝ$ and applications to minimal surfaces, Trans. Amer. Math. Soc., Volume 361 (2009) no. 12, pp. 6255-6282 | DOI | MR | Zbl

[9] Daniel, Benoît Minimal isometric immersions into $𝕊^{2} \times ℝ$ and $ℍ^{2} \times ℝ$ , Indiana Univ. Math. J., Volume 64 (2015) no. 5, pp. 1425-1445 | DOI | MR | Zbl

[10] Domínguez-Vázquez, Miguel; M. Manzano, José Isoparametric surfaces in $𝔼 (κ, τ)$ -spaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), Volume 22 (2021) no. 1, pp. 269-285 | DOI | MR | Zbl

[11] Eisenhart, Luthur Pfahler An Introduction to Differential Geometry, Princeton Mathematical Series, v. 3, Princeton University Press, Princeton, N. J., 1940, x+304 pages | MR | Zbl

[12] Espinar, José M.; Rosenberg, Harold Complete constant mean curvature surfaces in homogeneous spaces, Comment. Math. Helv., Volume 86 (2011) no. 3, pp. 659-674 | DOI | MR | Zbl

[13] Fernández, Isabel; Mira, Pablo Constant mean curvature surfaces in 3-dimensional Thurston geometries, Proceedings of the International Congress of Mathematicians. Volume II, Hindustan Book Agency, New Delhi (2010), pp. 830-861 | MR | Zbl

[14] Hirakawa, Shinya Constant Gaussian curvature surfaces with parallel mean curvature vector in two-dimensional complex space forms, Geom. Dedicata, Volume 118 (2006), pp. 229-244 | DOI | MR | Zbl

[15] Hoffman, David A. Surfaces of constant mean curvature in manifolds of constant curvature, J. Differential Geometry, Volume 8 (1973), pp. 161-176 | DOI | MR | Zbl

[16] Kenmotsu, Katsuei Minimal surfaces with constant curvature in $4$ -dimensional space forms, Proc. Amer. Math. Soc., Volume 89 (1983) no. 1, pp. 133-138 | DOI | MR | Zbl

[17] Kenmotsu, Katsuei; Masuda, Kyûya On minimal surfaces of constant curvature in two-dimensional complex space form, J. Reine Angew. Math., Volume 523 (2000), pp. 69-101 | DOI | MR | Zbl

[18] Lawson, H. Blaine Jr. Local rigidity theorems for minimal hypersurfaces, Ann. of Math. (2), Volume 89 (1969), pp. 187-197 | DOI | MR | Zbl

[19] Leite, Maria Luiza An elementary proof of the Abresch–Rosenberg theorem on constant mean curvature immersed surfaces in $𝕊^{2} \times ℝ$ and $ℍ^{2} \times ℝ$ , Q. J. Math., Volume 58 (2007) no. 4, pp. 479-487 | DOI | MR | Zbl

[20] Levi-Civita, Tullio Famiglie di superficie isoparametrische nell’ordinario spacio euclideo, Atti. Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Natur., Volume 26 (1937), pp. 355-362 | Zbl

[21] Peñafiel, Carlos Invariant surfaces in ${\tilde{P S L}}_{2} (ℝ, τ)$ and applications, Bull. Braz. Math. Soc. (N.S.), Volume 43 (2012) no. 4, pp. 545-578 | DOI | MR | Zbl

[22] Peñafiel, Carlos Screw motion surfaces in ${\tilde{P S L}}_{2} (ℝ, τ)$ , Asian J. Math., Volume 19 (2015) no. 2, pp. 265-280 | DOI | MR | Zbl

[23] Sa Earp, Ricardo; Toubiana, Eric Screw motion surfaces in $ℍ^{2} \times ℝ$ and $𝕊^{2} \times ℝ$ , Illinois J. Math., Volume 49 (2005) no. 4, pp. 1323-1362 | DOI | MR | Zbl

[24] Scott, Peter The geometries of $3$ -manifolds, Bull. London Math. Soc., Volume 15 (1983) no. 5, pp. 401-487 | DOI | MR | Zbl

[25] Torralbo, Francisco; Urbano, Francisco Surfaces with parallel mean curvature vector in $𝕊^{2} \times 𝕊^{2}$ and $ℍ^{2} \times ℍ^{2}$ , Trans. Amer. Math. Soc., Volume 364 (2012) no. 2, pp. 785-813 | DOI | MR | Zbl

[26] Torralbo, Francisco; Urbano, Francisco Minimal surfaces in $𝕊^{2} \times 𝕊^{2}$ , J. Geom. Anal., Volume 25 (2015) no. 2, pp. 1132-1156 | DOI | MR | Zbl

[27] Verpoort, Steven Hypersurfaces with a parallel higher fundamental form, J. Geom., Volume 105 (2014) no. 2, pp. 223-242 | DOI | MR | Zbl

[28] Wall, C. T. C. Geometric structures on compact complex analytic surfaces, Topology, Volume 25 (1986) no. 2, pp. 119-153 | DOI | MR | Zbl

Cited by Sources:

ANNALES DE L'INSTITUT FOURIER

ARTICLES TO APPEAR

BROWSE ISSUES

ABOUT THE JOURNAL

EDITORIAL BOARD

SUBMIT A PAPER

SUBSCRIPTION