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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 [{\it 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 $n-r$.

53C42Immersions (differential geometry)
53A10Minimal surfaces, surfaces with prescribed mean curvature
Full Text: DOI
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