Hypersurfaces in space forms with scalar curvature conditions (Q1431121)

The authors prove the following theorem: Let \(f\) be a minimal immersion of an \(n\)-dimensional complete Riemannian manifold \(M\) into the \((n+1)\)-dimensional unit sphere in the \((n+1)\)-dimensional Euclidean space. If \(f\) maps \(M\) into the upper closed hemisphere, then the isometric immersion is a total embedding provided that one of the following conditions is satisfied: (1) the normalized scalar curvature \(R\) of \(M\) is a constant; (2) \(\inf R> 1-(n+ 1)^2/(n- 1)^3\). As a corollary, the authors get the following theorem: If \(M\) is an \(n\)-dimensional \((n> 2)\) complete surface in \((n+1)\)-dimensional Euclidean space with normalized scalar curvature \(R\geq H^2 n(n- 2)/(n- 1)^2\), then \(M\) is a hyperplane or a sphere or a cylinder.