Index of Lagrangian submanifolds of \(\mathbb{C} \mathbb{P}^ n\) and the Laplacian of 1-forms (Q1312324)

Let \(\mathbb{C}\mathbb{P}^ n\) be complex projective space with the Fubini Study metric. Let \(\phi: M \to \mathbb{C}\mathbb{P}^ n\) be a minimal Lagrangian immersion of a compact manifold \(M\) and let \(\lambda_ i\) be the first two eigenvalues of the Laplacian acting on 1-forms. The author shows that \(\lambda_ 1 \leq (n - 1)/2\) with equality if and only if \(\phi\) is either the totally geodesic embedding of \(\mathbb{R}\mathbb{P}^ n(1/4)\) in \(\mathbb{C}\mathbb{P}^ n\) or is the totally geodesic immersion of \(S^ 2(1/4)\) in \(\mathbb{C}\mathbb{P}^ 2\). The author shows that \[ \{(n - 1)/2 - \lambda_ 1\}\{(n+1)/2 - \lambda_ 2\} + \lambda_ 1 \geq 2\int_ M \overline{\rho}/\text{vol}(M) \] where \(\overline{\rho}\) is the normalized scalar curvature of \(M\) with equality if and only if \(\phi\) is totally geodesic or if \(M\) is the generalized Clifford torus. The author uses these results to show that if \(M\) is an orientable surface, then the index of \(M\) is at least 2 with equality if and only if \(M\) is the Clifford torus. The author also shows the only Hamiltonian stable minimal Lagrangian torus in \(\mathbb{C}\mathbb{P}^ 2\) is the Clifford torus.