Global existence of generalized rotational hypersurfaces with prescribed mean curvature in Euclidean spaces, I

Abstract

We prove that for a given continuous function H(s), (−∞ < s < ∞), there exists a globally defined generating curve of a rotational hypersurface in a Euclidean space such that the mean curvature is H(s). We also prove a similar theorem for generalized rotational hypersurfaces of O(l+1) × O(m+1)-type. The key lemmas in this paper show the existence of solutions for singular initial value problems of ordinary differential equations satisfied using generating curves of those hypersurfaces.