Curvature pinching and eigenvalue rigidity for minimal submanifolds (Q1057524)

Simon's formula is a basic and useful tool in the study of some problems of global rigidity for minimal submanifolds immersed in kind Riemannian manifolds. In this paper a modified version of this formula is obtained, that even though this essentially coincides with the classic version, it allows a more controlled treatment and from new view points. We give sharp applications in two different contexts: First we obtain a complete solution to the pinching problem for the Ricci curvature of n-dimensional compact totally real sumanifolds immersed minimally into the n- dimensional complex projective space, and we characterize in this way six nice submanifolds of this family. Second we characterize the compact symmetric spaces of rank one among all compact Riemannian manifolds which admit a minimal immersion into a sphere by the first eigenfunctions as the extreme cases in several inequalities involving only the simplest spectral invariants: The first and the second eigenvalue of the Laplacian, the volume, the integral of the scalar curvature and the multiplicity of the first eigenvalue.