Complete linear Weingarten hypersurfaces immersed in the hyperbolic space. (English) Zbl 1303.53080

A hypersurface is called linear Weingarten if an equation of type \(R= aH+ b\) is satisfied for some constants \(a\), \(b\) where \(R\) denotes the scalar curvature and \(H\) is the mean curvature. The main result of this article is the following.
Theorem 1.1: A complete linear Weingarten hypersurfaces of the hyperbolic \((n+1)\)-space is either totally umbilical or isometric to a hyperbolic cylinder over a sphere of dimension \(1\) or \(n-1\) if a certain inequality is satisfied. This inequality assumes that the norm of the second fundamental form is not greater than a certain expression involving only the mean curvature \(H\) and the dimension \(n\). Moreover it is assumed that \(H\) attains its maximum on the manifold.


53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53C20 Global Riemannian geometry, including pinching
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
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