Integral Inequalities for Compact Hypersurfaces with Constant Scalar Curvature in the Euclidean Sphere

摘要

We study the rigidity of compact-oriented hypersurfaces with constant scalar curvature isometrically immersed into the unit Euclidean sphere Sn+1. In particular, we establish a sharp integral inequality for the behavior of the norm of the total umbilicity tensor, equality characterizing the totally umbilical hypersurfaces, and a certain family of standard tori of the form S-1(root 1 - r(2)) x Sn-1(r). Moreover, under an appropriate constraint on the total umbilicity tensor, we are able to extend this result for any integer k, with 2 <= k <= n - 1, equality characterizing the totally umbilical hypersurfaces and a certain family of standard product of spheres of the form S-k (root 1 - r(2)) x Sn-k(r).