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Hypersurfaces with null higher order mean curvature. (English) Zbl 1210.53057
The constancy condition of higher elementary symmetric functions of the principal curvatures (= higher mean curvatures H r ) is an interesting problem in the global theory of hypersurfaces. For the compact case and a non-vanishing constant see [R. Walter, Math. Ann. 270, 125–145 (1985; Zbl 0536.53054)]. In the present paper, the authors deal with the case of vanishing higher mean curvature of complete and orientable hypersurfaces of dimension n in space forms. They make the assumption that an open part does not consist of totally geodesic hypersurfaces. Another assumption is that H r+1 vanishes identically and H r never vanishes. One of the results states that, under these assumptions, the manifold is foliated by complete totally geodesic submanifolds of dimension n-r.
MSC:
53C42Immersions (differential geometry)
53A10Minimal surfaces, surfaces with prescribed mean curvature
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