Simons Type Equation in $\mathbb{S}^{2}\times \mathbb{R}$ and $\mathbb{H}^2\times \mathbb{R}$ and Applications

Batista da Silva, Márcio Henrique

doi:10.5802/aif.2641

Simons Type Equation in

𝕊^{2} \times ℝ

and

ℍ^{2} \times ℝ

and Applications
[Les équations de type Simons dans

𝕊^{2} \times ℝ

ℍ^{2} \times ℝ

et applications]

Batista da Silva, Márcio Henrique ¹

¹ Universidade Federal de Alagoas Instituto de Matemática CEP: 57072-900 Maceió - Alagoas (Brazil)

Annales de l'Institut Fourier, Tome 61 (2011) no. 4, pp. 1299-1322.

Résumé
Abstract

Soit $Σ^{2}$ une surface immergée dans $M^{2} (c) \times ℝ$ avec une courbure moyenne constante. Nous considérons l’opérateur de Weingarten à trace nulle $φ$ associé à la seconde forme fondamentale de la surface et nous introduisons un tenseur $S$ , liés à la forme quadratique de Abresch-Rosenberg. Nous établissons les équations de type Simons pour $φ$ et $S$ . En utilisant ces équations, nous caractérisons les immersions pour lesquelles $| φ |$ ou $| S |$ sont bornés.

Let $Σ^{2}$ be an immersed surface in $M^{2} (c) \times ℝ$ with constant mean curvature. We consider the traceless Weingarten operator $φ$ associated to the second fundamental form of the surface, and we introduce a tensor $S$ , related to the Abresch-Rosenberg quadratic differential form. We establish equations of Simons type for both $φ$ and $S$ . By using these equations, we characterize some immersions for which $| φ |$ or $| S |$ is appropriately bounded.

MR Zbl

DOI : 10.5802/aif.2641

Classification : 53A10, 53C42
Keywords: Surface with constant mean curvature, Simons type equation, Codazzi’s equation
Mot clés : surface à courbure moyenne constante, équation type Simons, équation de Codazzi

Affiliations des auteurs :

Batista da Silva, Márcio Henrique ¹

¹ Universidade Federal de Alagoas Instituto de Matemática CEP: 57072-900 Maceió - Alagoas (Brazil)

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     title = {Simons {Type} {Equation} in $\mathbb{S}^{2}\times \mathbb{R}$ and $\mathbb{H}^2\times \mathbb{R}$ and {Applications}},
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Batista da Silva, Márcio Henrique. Simons Type Equation in $\mathbb{S}^{2}\times \mathbb{R}$ and $\mathbb{H}^2\times \mathbb{R}$ and Applications. Annales de l'Institut Fourier, Tome 61 (2011) no. 4, pp. 1299-1322. doi : 10.5802/aif.2641. http://www.numdam.org/articles/10.5802/aif.2641/

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