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

Abstract

We prove that for a given continuous function $H(s)$, $(-\infty$ < $s$ < $\infty)$, 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 $O(l+1)\times O(m+1)$-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.