Rotational hypersurfaces of periodic mean curvature

Abstract

In this paper we discuss rotational hypersurfaces in R n and more specifically rotational hypersurfaces with periodic mean curvature function. We show that, for a given real analytic function H ( s ) on R , every rotational hypersurface M in R n with mean curvature H ( s ) can be extended infinitely in the sense that all coordinate functions of the generating curve of M are defined on all of R as well. For rotational hypersurfaces with periodic mean curvature we present a criterion characterizing the periodicity of such hypersurfaces in terms of their mean curvature function. We also discuss a method to produce families of periodic rotational hypersurfaces where each member of the family has the same mean curvature function. In fact, given any closed planar curve with curvature κ, we prove that there is a family of periodic rotational hypersurfaces such that the mean curvature of each element of the family is explicitly determined by κ. Delaunay's famous result for surfaces of revolution with constant mean curvature is included here as the case where n = 3 and κ is constant.