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

TL;DR

This paper proves the infinite extendability of rotational hypersurfaces with prescribed mean curvature in Euclidean spaces, generalizing classical results and analyzing solutions of associated differential equations.

Contribution

It extends the theory of rotational hypersurfaces with prescribed mean curvature to more general types, including O(l+1) x O(m+1)-type hypersurfaces, using novel solution existence lemmas.

Findings

Unique infinite extension of hypersurfaces with prescribed mean curvature.

Generalization of classical results by Euler and Delaunay.

Existence of solutions for singular initial value problems.

Abstract

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