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

Abstract

We study the rigidity of compact-oriented hypersurfaces with constant scalar curvature isometrically immersed into the unit Euclidean sphere ${\mathbb {S}}^{n+1}$. 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 ${\mathbb {S}}^1(\sqrt{1-r^2})\times {\mathbb {S}}^{n-1}(r)$. Moreover, under an appropriate constraint on the total umbilicity tensor, we are able to extend this result for any integer k, with $2\le k\le n-1$, equality characterizing the totally umbilical hypersurfaces and a certain family of standard product of spheres of the form ${\mathbb {S}}^k(\sqrt{1-r^2})\times {\mathbb {S}}^{n-k}(r)$.