Rotational hypersurfaces of prescribed mean curvature

Abstract

We use a phase space analysis to give some classification results for rotational hypersurfaces in ${\mathbb{R}}^{n+1}$ whose mean curvature is given as a prescribed function of its Gauss map. For the case where the prescribed function is an even function in ${\mathbb{S}}^n$, we show that a Delaunay-type classification holds for this class of hypersurfaces. We also exhibit examples showing that the behavior of rotational hypersurfaces of prescribed (non-constant) mean curvature is much richer than in the constant mean curvature case.

23 pages, 30 figures. The results of this paper were originally contained in our previous posting arXiv:1802.08146 which, because of its length and following the editor's suggestion, has now been divided into two papers; the present paper is one of them