On the isolation phenomena of locally conformally flat manifolds with constant scalar curvature - Submanifolds versions

In this paper, from the viewpoint of submanifold theory, we study the isolation phenomena of Riemannian manifolds with constant scalar curvature and vanishing Weyl conformal curvature tensor. Firstly, for any locally strongly convex affine hyperspheres in an (n+1)-dimensional equiaffine space Rn+1 with constant scalar curvature, we prove an inequality involving the traceless Ricci tensor, the Pick invariant and the scalar curvature. The inequality is optimal and we can further completely classify the affine hyperspheres which realize the equality case of the inequality. Secondly, and analogously, for Lagrangian minimal submanifolds of the complex projective space CPn equipped with the Fubini-Study metric, under the condition that the Weyl conformal curvature tensor vanishes, we establish a similar but reverse inequality involving the traceless Ricci tensor, the scalar curvature and the squared norm of the second fundamental form. The inequality is also optimal and we can further completely classify the submanifolds which realize the equality case of the inequality.