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We prove that any piece of a rotational hypersurface with prescribed meancurvature function in a Euclidean space can be uniquely extended infinitely,which generalizes the results by Euler and Delaunay for surfaces of revolutionwith constant mean curvautre. Next, we prove the same kind of theorem forgeneralized rotational hypersurfaces of O(l+1) x O(m+1)-type. The key lemmas inthis paper show the existence of solutions for singular initial value problemswhich arise from the analysis of ordinary differential equations of generatingcurves of those hypersurfaces.

On the global existence of generalized rotational hypersurfaces with prescribed mean curvature in the Euclidean spaces. I