Abstract
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 ≥ 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.
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Communicated by D. Alekseevsky.
A. Brasil Jr. was partially supported by CNPq, Brazil. A. G. Colares was partially supported by FUNCAP, Brazil. O. Palmas was partially supported by CNPq, Brazil and DGAPA-UNAM, México, under Project IN118508.
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Brasil, A., Colares, A.G. & Palmas, O. Complete hypersurfaces with constant scalar curvature in spheres. Monatsh Math 161, 369–380 (2010). https://doi.org/10.1007/s00605-009-0128-9
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DOI: https://doi.org/10.1007/s00605-009-0128-9