Complete rotation hypersurfaces with Hk constant in space forms

O(p + 1) × O(q + 1)-invariant (r – 1)-minimal hypersurfaces in Euclidean space

“…A classification of rotational hypersurfaces with zero scalar curvature and O(n)-invariant, as defined in [6], of a space form was made by Leite (see [12]) and later generalized by Palmas for H r constant, see [13].…”

New examples ofWr-minimal hypersurfaces in a sphere

“…From [Palmas 1999] and [Wei 2007], we know that there exist many compact immersed k-minimal rotational hypersurfaces in a unit sphere S n+1 (1) for 1 ≤ k ≤ n − 1.…”

“…Putting (2-4) and (2-5) into (2-6) gives another theorem: Palmas 1999]. The rotational hypersurface M n in S n+1 (1) is k-minimal with k < n if and only if f satisfies the differential equation…”

Complete hypersurfaces in R 2 n +2 with constant negative 2 n -th curvature

“…As this universe is still large, we restrict further to those such hypersurfaces which are invariant under a given group of isometries of the euclidean space. Following the classification of the low cohomogeneity isometry groups given by W. Y. Hsiang and H. B. Lawson in [2], the author studied rotational (i.e., SO(n)-invariant) hypersurfaces in space forms in [5], and concluded, on one hand, that for each r odd and each σ < 0 there exists a 1-parameter family of complete rotational hypersurfaces in R n+1 with σ r = σ ; and, on the other hand, that there are no complete rotational hypersurfaces with constant negative σ r for r even. It should be mentioned that these results generalize those obtained by M. L. Leite in [3] and J. Hounie and Leite in [1].…”