Hipersuperfícies completas com k-ésima função simétrica nula na esfera unitária

Abstract

n this work we will study hypersurfaces M^n of the unit sphere S^{n+1} complete connected with two distinct principal curvatures one of which of multiplicity n-1 and having k-th null curvature function. Under these conditions we will prove that the torus Clifford's is the only hypersurface that satisfies S greater than or equal to n(k^2-2k+n)}/{k(nk)}=c(n,k) where S represents the square of the norm of the second fundamental form Beyond from this we will show that in the compact case the integral over M of S is less than or equal to c(n,k)vol(M) with equality occurring only in the Clifford torus