Global existence of generalized rotational hypersurfaces with prescribed mean curvature in Euclidean spaces. I. (English) Zbl 1325.53017

Summary: 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.


53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
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