A new characterization of submanifolds with parallel mean curvature vector in \(S^{n+p}\) (Q1884367)

Let \((M^n,g)\) be a compact Riemannian submanifold of \(S^{n+p}\) with parallel mean curvature vector \(h\). Let \(H\) denote the length of \(h\) and \(A\) the second fundamental form of \(M^n\). Put \(\Phi=A-gh\) and \(\Phi_h(X,Y)=\langle\Phi(X,Y), h\rangle\) for \(X, Y\) tangent to \(M^n\). Define the Schrödinger-type operator \(L\) by \(L=-\Delta-B| \Phi| ^2-C| \Phi_h| \), where \(C=n(n-2)/\sqrt{n(n-1)}\) and \(B\) is equal to \(2-1/p\) for \(h=0\) or \(p=1\) and to \(2-1/(p-1)\) for \(p>1\). The main result of this paper states that if the first eigenvalue \(\mu_1\) of \(L\) satisfies \(\mu_1\geq -n(1+H^n)\), then \(\mu_1=0\) or \(\mu_1=-n(1+H^2)\). Moreover, \(\mu_1=0\) if and only if \(M^n\) is totally umbilic and \(\mu_1=-n(1+H^2)\) if and only if \(M^n\) is either the Veronese surface or the Clifford torus.