Abstract:

Let $M$ be a Willmore (or extremal) hypersurface in $S^{n+1}$ with the same squared length of the second fundamental form of Willmore torus $W_{m,n-m}$ (or Clifford torus $C_{m,n-m}$). In this article the authors proved that if $Spec^p(M)=Spec^p(W_{m,n-m})$ (or $Spec^p(M)=Spec^p(C_{m,n-m})$) for $p=0,1,2$, then $M$ is $W_{m,n-m}$ (or $C_{m,m}$).

Key words: Spectrum, Laplace operator, Extremal hypersurface, Willmore hypersurface, The second fundamental form