Journal of the Mathematical Society of Japan

Complete linear Weingarten hypersurfaces immersed in the hyperbolic space

Henrique Fernandes DE LIMA

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In this paper, we apply the Hopf's strong maximum principle in order to obtain a suitable characterization of the complete linear Weingarten hypersurfaces immersed in the hyperbolic space $\mathbb H^{n+1}$. Under the assumption that the mean curvature attains its maximum and supposing an appropriated restriction on the norm of the second fundamental form, we prove that such a hypersurface must be either totally umbilical or isometric to a hyperbolic cylinder of $\mathbb H^{n+1}$.

Article information

J. Math. Soc. Japan, Volume 66, Number 2 (2014), 415-423.

First available in Project Euclid: 23 April 2014

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Zentralblatt MATH identifier

Primary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]
Secondary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42] 53C20: Global Riemannian geometry, including pinching [See also 31C12, 58B20] 53C50: Lorentz manifolds, manifolds with indefinite metrics

hyperbolic space linear Weingarten hypersurfaces totally umbilical hypersurfaces hyperbolic cylinders


DE LIMA, Henrique Fernandes. Complete linear Weingarten hypersurfaces immersed in the hyperbolic space. J. Math. Soc. Japan 66 (2014), no. 2, 415--423. doi:10.2969/jmsj/06620415.

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