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 q (resp. RP n ) be the complex hyperquadric (resp. the totally geodesic real projective space) in the complex projective space CP n of constant holomorphic sectional curvature 4l. We prove that if the Ricci and some n À 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 volumePavolumeM % volumeqavolumeCP n (resp. % volumeRP n avolumeCP n ), then there is a holomorphic isometry between M and CP n taking P isometrically onto q (resp. RP 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 À 1-Ricci curvatures bounded from below by those of the tube RP n r of radius r around RP n in CP n and have the first Dirichlet eigenvalue not lower than that of RP n r .1. Introduction. Let us consider a family of pairs PY M, where M is a compact riemannian manifold and P is a compact riemannian submanifold of M. In the last years some work has been done to get information on the topology and geometry of the pair PY M from bounds on extrinsic curvatures of P and intrinsic curvatures of M (cfr. [18], [19], [3], [8], [9], [12], [4]). In particular, in the last four references they give comparison theorems for the quotient volumePavolumeM and the first Dirichlet eigenvalue of a tube around P. When M is a Kähler manifold, some results have been got by some people and the author (cfr. [13], [5], [15]). In [16], V. Palmer and the author have improved some estimates given in [5] and [15]for the quotient volumePavolumeM (and other riemannian invariants) when P is a complex hypersurface with the norm of the second fundamental form strictly positive. We have also got bounds of these invariants when P is a lagrangian submanifold. However, unlike in the preceeding results about pairs, we were not able to prove that the equality with the bound of the last riemannian invariants should characterize the pairs (complex hyperquadric, complex projective space) and (real projective space, complex projective space) where the bounds are attained (only partial results in that direction were obtained).
