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LINEAR WEINGARTEN HYPERSURFACES WITH BOUNDED MEAN CURVATURE IN THE HYPERBOLIC SPACE

Part of: Global differential geometry

Published online by Cambridge University Press:  17 December 2014

CÍCERO P. AQUINO ,
HENRIQUE F. DE LIMA  and
MARCO ANTONIO L. VELÁSQUEZ
CÍCERO P. AQUINO
Affiliation:
Departamento de Matemática, Universidade Federal do Piauí, 64.049-550 Teresina, Piauí, Brazil E-mail: cicero.aquino@ufpi.edu.br
HENRIQUE F. DE LIMA
Affiliation:
Departamento de Matemática, Universidade Federal de Campina Grande, 58.429-970 Campina Grande, Paraíba, Brazil E-mail: henrique@dme.ufcg.edu.br
MARCO ANTONIO L. VELÁSQUEZ
Affiliation:
Departamento de Matemática, Universidade Federal de Campina Grande, 58.429-970 Campina Grande, Paraíba, Brazil E-mail: marco.velasquez@pq.cnpq.br
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Abstract

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We apply appropriate maximum principles in order to obtain characterization results concerning complete linear Weingarten hypersurfaces with bounded mean curvature in the hyperbolic space. By supposing a suitable restriction on the norm of the traceless part of the second fundamental form, we show that such a hypersurface must be either totally umbilical or isometric to a hyperbolic cylinder, when its scalar curvature is positive, or to a spherical cylinder, when its scalar curvature is negative. Related to the compact case, we also establish a rigidity result.

MSC classification

Primary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.)
Secondary: 53C40: Global submanifolds
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 57 , Issue 3 , September 2015 , pp. 653 - 663
Copyright
Copyright © Glasgow Mathematical Journal Trust 2014 

References

REFERENCES

1.Alencar, H. and do Carmo, M., Hypersurfaces with constant mean curvature in spheres, Proc. Am. Math. Soc. 120 (1994), 1223–1229.CrossRefGoogle Scholar
2.Alías, L. J. and García-Martínez, S. C., On the scalar curvature of constant mean curvature hypersurfaces in space forms J. Math. Anal. Appl. 363 (2010), 579–587.CrossRefGoogle Scholar
3.Alías, L. J., García-Martínez, S. C. and Rigoli, M., A maximum principle for hypersurfaces with constant scalar curvature and applications, Ann. Glob. Anal. Geom. 41 (2012), 307–320.CrossRefGoogle Scholar
4.Aquino, C. P. and de Lima, H. F., On the geometry of linearWeingarten hypersurfaces in the hyperbolic space, Monatsh. Math. 171 (2013), 259–268.CrossRefGoogle Scholar
5.Barros, A., Silva, J. and Sousa, P., Rotational Linear Weingarten surfaces into the Euclidean sphere, Israel J. Math. 192 (2012), 819–830.CrossRefGoogle Scholar
6.Barros, A., Silva, J. and Sousa, P., Rotational linear Weingarten hypersurfaces into the Euclidean sphere $\mathbb{S}$n+1, Adv. Geom., to appear.Google Scholar
7.Brasil, A. Jr., Colares, A. G. and Palmas, O., Complete hypersurfaces with constant scalar curvature in spheres, Monatsh. Math. 161 (2010), 369–380.CrossRefGoogle Scholar
8.Caminha, A., The geometry of closed conformal vector fields on Riemannian spaces Bull. Braz. Math. Soc. 42 (2011), 277–300.CrossRefGoogle Scholar
9.Cartan, É., Familles de surfaces isoparamétriques dans les espaces à courbure constante, Ann. Mat. Pura Appl. 17 (1938), 177–191.CrossRefGoogle Scholar
10.Cheng, S. Y. and Yau, S. T., Hypersurfaces with constant scalar curvature Math. Ann. 225 (1977), 195–204.CrossRefGoogle Scholar
11.Li, H., Hypersurfaces with constant scalar curvature in space forms Math. Ann. 305 (1996), 665–672.Google Scholar
12.Li, H., Suh, Y. J. and Wei, G., Linear Weingarten hypersurfaces in a unit sphere, Bull. Korean Math. Soc. 46 (2009), 321–329.CrossRefGoogle Scholar
13.López, R., Rotational linear Weingarten surfaces of hyperbolic type Israel J. Math. 167 (2008), 283–301.CrossRefGoogle Scholar
14.Okumura, M., Hypersurfaces and a pinching problem on the second fundamental tensor Am. J. Math. 96 (1974), 207–213.CrossRefGoogle Scholar
15.Omori, H., Isometric immersions of Riemannian manifolds J. Math. Soc. Japan 19 (1967), 205–214.CrossRefGoogle Scholar
16.Pigola, S., Rigoli, M. and Setti, A. G., Maximum principles on Riemannian manifolds and applications, Mem. Am. Math. Soc. 822 (2005), 1–95.Google Scholar
17.Ryan, P. J., Hypersurfaces with parallel Ricci tensor Osaka J. Math. 8 (1971), 251–259.Google Scholar
18.Shu, S., Complete hypersurfaces with constant scalar curvature in a hyperbolic space Balkan J. Geom. Appl. 12 (2007), 107–115.Google Scholar
19.Shu, S., Linear Weingarten hypersurfaces in a real space form Glasgow Math. J. 52 (2010), 635–648.CrossRefGoogle Scholar
20.Yau, S. T., Harmonic functions on complete Riemannian manifolds Commun. Pure Appl. Math. 28 (1975), 201–228.CrossRefGoogle Scholar
21.Yau, S.T., Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry Indiana Univ. Math. J. 25 (1976), 659–670.CrossRefGoogle Scholar