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

De Lima, Henrique Fernandes

J. Math. Soc. Japan 66, No. 2, 415-423 (2014).

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.

Reviewer: Wolfgang Kühnel (Stuttgart)

MSC:

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

Keywords:

maximum principle; hyperbolic cylinder; Cheng-Yau operator

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