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Hypersurfaces of constant curvature in space forms. (English) Zbl 0787.53046
Let \(M^ n\) be a compact embedded hypersurface in the space form \(N^{m+1}(c)\) (\(c=0\) or \(c=-1\)) with boundary contained in a totally geodesic hyperplane, having one symmetric function \(S_{r+1}\) of principal curvatures a positive constant. In this interesting paper the author obtains height estimates and derives some applications of them. As an example we note Ph. Hartman’s theorem [Trans. Am. Math. Soc. 245, 363-374 (1978; Zbl 0412.53027)]: Let \(M\) be a complete embedded hypersurface of \(R^{m+1}\) with \(S_{r+1}\) a positive constant. If \(M\) has nonnegative sectional curvature then \(M\) is isometric to \(S^ p \times R^ \ell\), \(S^ p\) a round sphere. The proofs depend on obtaining some elliptic equations on \(M\), by calculating the first variation of symmetric functions of principal curvatures.

53C40 Global submanifolds