A rigidity theorem for the pair ${\cal q}{\Bbb C} P^n$ (complex hyperquadric, complex projective space)

Abstract.

Given a compact Kähler manifold M of real dimension 2n, let P be either a compact complex hypersurface of M or a compact totally real submanifold of dimension n. Let $\cal q$ (resp. ${\Bbb R} P^n$) be the complex hyperquadric (resp. the totally geodesic real projective space) in the complex projective space ${\Bbb C} P^n$ of constant holomorphic sectional curvature 4$\lambda$. We prove that if the Ricci and some (n-1)-Ricci curvatures of M (and, when P is complex, the mean absolute curvature of P) are bounded from below by some special constants and volume (P) / volume (M) $\leq$ volume ($\cal q$)/ volume $({\Bbb C} P^n)$ (resp. $\leq$ volume $({\Bbb R} P^n)$ / volume $({\Bbb C} P^n)$), then there is a holomorphic isometry between M and ${\Bbb C} P^n$ taking P isometrically onto $\cal q$ (resp. ${\Bbb R} P^n$). We also classify the Kähler manifolds with boundary which are tubes of radius r around totally real and totally geodesic submanifolds of half dimension, have the holomorphic sectional and some (n-1)-Ricci curvatures bounded from below by those of the tube ${\Bbb R} P^n_r$ of radius r around ${\Bbb R} P^n$ in ${\Bbb C} P^n$ and have the first Dirichlet eigenvalue not lower than that of ${\Bbb R} P^n_r$.