L-k-2-TYPE HYPERSURFACES IN HYPERBOLIC SPACES

摘要

In this article, we study L-k-finite-type hypersurfaces M-n of a hyperbolic space Hn+1 subset of R-1(n+2) for k >= 1. In the 3-dimensional case, we obtain the following classification result. Let psi: M-3 -> H-4 subset of R-1(5) be an orientable hypersurface with constant k-th mean curvature H-k, which is not totally umbilical. Then M-3 is of L-k-2-type if and only if M-3 is an open portion of a standard Riemannian product H-1(r1) x S-2(r2) or H-2(r1) x S-1(r2), with -r(1)(2) + r(2)(2) = -1. In the n-dimensional case, we show that a hypersurface M-n subset of Hn+1, with constant k-th mean curvature H-k and having at most two distinct principal curvatures, is of L-k-2-type if and only if M-n is an open portion of a Riemannian product H-m(r(1)) x Sn-m (r(2)), with -r(1)(2) + r(2)(2) = -1, for some integer m is an element of{1, ... , n-1}. In the case k = n-1 we drop the condition on the principal curvatures of the hypersurfaceMn, and prove that ifMn subset of Hn+1 is an orientableHn-1-hypersurface of Ln-1-2-type then its Gauss-Kronecker curvature H-n is a nonzero constant.