New Spectral Characterizations of Extremal Hypersurfaces

Let M be a closed extremal hypersurface in Sn+1 with the same mean curvature of the Willmore torus Wm, n-m. We proved that if Specp(M) = Specp(Wm, n-m) for p = 0,1,2, then M is Wm, m.