The generalized Lu rigidity theorem for submanifolds with parallel mean curvature (Q1692181)

Let \(M\) be an \(n\)-dimensional oriented compact submanifold with parallel mean curvature in the unit sphere \(S^{n+p}\). Let \(\lambda_2\) the second largest eigenvalue of the fundamental matrix of \(M\) and let \[ \alpha(n,H)=n +\frac{n^3 H^2}{2(n-1)} -\frac{n(n-2)H}{2(n-1)}\sqrt{n^2 H^2 +4(n-1)} \] where \(H\) is the mean curvature of \(M\). Let \(S\) be the squared length of the second fundamental form \(h\) of \(M\). If \(S +\lambda_2 \leq \alpha(n,H)\), then the authors obtain a rigidity theorem for \(M\), which generalizes Lu's optimal rigidity theorem for minimal submanifolds in the unit sphere \(S^{n+p}\).