Complete hypersurfaces with constant scalar curvature in spheres

Abstract

To a given immersion i : M-n -> Sn+1 with constant scalar curvature R, we associate the supremum of the squared norm of the second fundamental form sup vertical bar A vertical bar(2). We prove the existence of a constant C (n) (R) depending on R and n so that R a parts per thousand yen 1 and sup vertical bar A vertical bar(2) = C (n) (R) imply that the hypersurface is a H(r)-torus S-1(root 1-r(2)) x Sn-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 Sn+1 with constant scalar curvature R and sup vertical bar A vertical bar(2) = C, answering questions raised by Q. M. Cheng.