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

Batista da Silva, Márcio Henrique

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

Annales de l'Institut Fourier, Tome 61 (2011), p. 1299-1322 / Harvested from Numdam

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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.

Publié le : 2011-01-01

MR 2951494 | Zbl 1242.53066

DOI : https://doi.org/10.5802/aif.2641
Classification: 53A10, 53C42
Mots clés: surface à courbure moyenne constante, équation type Simons, équation de Codazzi

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     author = {Batista da Silva, M\'arcio Henrique},
     title = {Simons Type Equation in $\mathbb{S}^{2}\times \mathbb{R}$ and $\mathbb{H}^2\times \mathbb{R}$ and Applications},
     journal = {Annales de l'Institut Fourier},
     volume = {61},
     year = {2011},
     pages = {1299-1322},
     doi = {10.5802/aif.2641},
     zbl = {1242.53066},
     mrnumber = {2951494},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2011__61_4_1299_0}
}

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) pp. 1299-1322. doi : 10.5802/aif.2641. http://gdmltest.u-ga.fr/item/AIF_2011__61_4_1299_0/

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