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Rotational hypersurfaces of prescribed mean curvature. (English) Zbl 1431.53009

Summary: We use a phase space analysis to give some classification results for rotational hypersurfaces in \(\mathbb{R}^{n + 1}\) whose mean curvature is given as a prescribed function of its Gauss map. For the case where the prescribed function is an even function in \(\mathbb{S}^n\), we show that a Delaunay-type classification holds for this class of hypersurfaces. We also exhibit examples showing that the behavior of rotational hypersurfaces of prescribed (non-constant) mean curvature is much richer than in the constant mean curvature case.

MSC:

53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C40 Ordinary differential equations and systems on manifolds
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