On spectral geometry of Kähler submanifolds (Q797848)

The main result is the following: Let \(M^ n\) be a compact Kähler submanifold, of complex dimension n, immersed in the complex projective space, \(CP^ m\), with the Fubini-Study metric of constant holomorphic sectional curvature 1. Let vol(M) and r be the volume and the scalar curvature of M, and \(\lambda_ 1\), \(\lambda_ 2\) the first and the second eigenvalue of the Laplacian of M. Then we have the following inequality: \[ n[n+1+(n+1-\lambda_ 1)(n+1-\lambda_ 2)]vol(M)\geq\int_{M}r. \] If equality holds then M is Einstein and (if the immersion is full) a certain tensor in the normal bundle of M is proportional to the metric. We know six different complex submanifolds of \(CP^ m\) for which the equality in the formula holds. As applications we give different spectral characterizations of these manifolds. Further results in this direction have been obtained by \textit{S. Udagawa} [''Spectral geometry of Kähler submanifolds in the complex projective space'', to appear in J. Math. Soc. Japan].