no. 4
Simons Type Equation in đť•Š 2 Ă—â„ť and â„Ť 2 Ă—â„ť and Applications
Batista da Silva, Márcio Henrique1
1 Universidade Federal de Alagoas Instituto de Matemática CEP: 57072-900 Maceió - Alagoas (Brazil)
Annales de l'Institut Fourier, Volume 61 (2011) no. 4, pp. 1299-1322.

Let Σ 2 be an immersed surface in M 2 (c)×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.

Soit Σ 2 une surface immergée dans M 2 (c)×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.

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

Batista da Silva, Márcio Henrique 1

1 Universidade Federal de Alagoas Instituto de Matemática CEP: 57072-900 Maceió - Alagoas (Brazil) 
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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, Volume 61 (2011) no. 4, pp. 1299-1322. doi : 10.5802/aif.2641. https://aif.centre-mersenne.org/articles/10.5802/aif.2641/

[1] Abresch, U.; Rosenberg, H. A Hopf differential for constant mean curvature surfaces in đť•Š 2 Ă—â„ť and â„Ť 2 Ă—â„ť, Acta Math., Volume 193 (2004), pp. 141-174 | DOI | MR | Zbl

[2] Alencar, H.; do Carmo, M. Hypersurfaces with constant mean curvature in Spheres, Proc. of the AMS, Volume 120 (1994), pp. 1223-1229 | DOI | MR | Zbl

[3] BĂ©rard, P. Simons Equation Revisited, Anais Acad. Brasil. CiĂŞncias, Volume 66 (1994), pp. 397-403 | Zbl

[4] Cheng, S.Y. Eigenfunctions and Nodal Sets, Commentarii Math. Helv., Volume 51 (1976), pp. 43-55 | DOI | MR | Zbl

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

[6] Hsiang, W.Y.; Hsiang, W.T. On the uniqueness of isoperimetric solutions and imbedded soap bubbles in noncompact symmetric spaces I, Invent. Math, Volume 98 (1989), pp. 39-58 | DOI | MR | Zbl

[7] Pedrosa, R.; Ritoré, M. Isoperimetric domains in the Riemannian product of a cicle with a simply connected space form and applications to free boundary problems, Indiana Univ. Math. J., Volume 48 (1999), pp. 1357-1394 | DOI | MR | Zbl

[8] Simons, J. Minimal varieties in Riemannian manifolds, Indiana Univ. Math. J., Volume 88 (1968), pp. 62-105 | MR | Zbl

[9] Souam, R.; Toubiana, E. Totally umbilic surfaces in homogeneous 3-manifolds (To appear in Comment. Math. Helv.) | Zbl

[10] Yau, S.T. Harmonic functions on complete Riemannian manifolds, Comm.Pure and Appl. Math., Volume 28 (1975), pp. 201-228 | DOI | MR | Zbl

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