Complete hypersurfaces with constant scalar curvature in spheres. (English) Zbl 1201.53068
Summary: To a given immersion \({i:M^n\to \mathbb S^{n+1}}\) with constant scalar curvature \(R\), we associate the supremum of the squared norm of the second fundamental form sup \(|A|^{2}\). We prove the existence of a constant \(C _{n }(R)\) depending on \(R\) and \(n\) so that \(R \geq 1\) and sup \(|A|^{2} = C _{n }(R)\) imply that the hypersurface is a \(H(r)\)-torus \({\mathbb S^1(\sqrt{1-r^2})\times\mathbb S^{n-1} (r)}\). For \(R > (n - 2)/n\) we use rotation hypersurfaces to show that for each value \(C > C _{n }(R)\) there is a complete hypersurface in \({\mathbb S^{n+1}}\) with constant scalar curvature \(R\) and sup \(|A|^{2} = C\), answering questions raised by Q. M. Cheng.
MSC:
| 53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |
| 53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |
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