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

2013/07/11 by Katsuei Kenmotsu, Kenmotsu, Katsuei, Takeyuki Nagasawa +1
Mathematics · Physics and Astronomy · #53C42 (primary) 34B16 (secondary) #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.DG #msc:34B16 #msc:53C42

paper · pdf · doi:10.48550/arxiv.1307.3068

35 pages

arxiv created 2013/07/11 · openalex publication_date 2013/07/11 · arxiv updated 2013/07/12 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

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.

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