Vol. 261, No. 1, 2013

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A new characterization of complete linear Weingarten hypersurfaces in real space forms

Cícero P. Aquino, Henrique F. de Lima and Marco A. L. Velásquez

Vol. 261 (2013), No. 1, 33–43

We apply the Hopf’s strong maximum principle in order to obtain a suitable characterization of the complete linear Weingarten hypersurfaces immersed in a real space form cn+1 of constant sectional curvature c. Under the assumption that the mean curvature attains its maximum and supposing an appropriated restriction on the norm of the traceless part of the second fundamental form, we prove that such a hypersurface must be either totally umbilical or isometric to a Clifford torus, if c = 1, a circular cylinder, if c = 0, or a hyperbolic cylinder, if c = 1.

space forms, linear Weingarten hypersurfaces, totally umbilical hypersurfaces, Clifford torus, circular cylinder, hyperbolic cylinder
Mathematical Subject Classification 2010
Primary: 53C42
Secondary: 53A10, 53C20, 53C50
Received: 29 June 2012
Revised: 24 August 2012
Accepted: 4 September 2012
Published: 28 February 2013
Cícero P. Aquino
Departamento de Matemática
Universidade Federal do Piauí
64049-550 Teresina, Piauí
Henrique F. de Lima
Departamento de Matemática e Estatística
Universidade Federal de Campina Grande
58429-970 Campina Grande, Paraíba
Marco A. L. Velásquez
Departamento de Matemática e Estatística
Universidade Federal de Campina Grande
58.429-970 Campina Grande, Paraíba