## 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.

### MSC:

 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

### Keywords:

mean curvature; generalized rotational hypersurfaces
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### References:

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