6533b7cefe1ef96bd1256eab

RESEARCH PRODUCT

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

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Given a compact Kahler 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 Kahler 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 \({\Bbb R} P^n_r\).