Hypersurfaces with constant higher order mean curvature in Euclidean space

Abstract

In this paper we derive sharp estimates for the infimum and for the supremum of the squared norm of the second fundamental form of complete oriented hypersurfaces of Euclidean space with constant higher order mean curvature and having two principal curvatures, one of them simple. Besides, we characterize those hypersurfaces for which any of these bounds is attained. Our results will be an application of a purely geometric result on the principal curvatures of the hypersurface, the so called principal curvature theorem, given by Smyth and Xavier (Invent Math 90:443–450, 1987).