  We consider the evolution by mean curvature flow of Lagrangian submanifolds
of the complex projective space CP^n. We prove that, if the initial value
satisfies a suitable pinching condition, then the flow exists for all times and
the manifold converges to a totally geodesic submanifold. As a corollary, we
obtain that a Lagrangian submanifold satisfying our pinching condition is
diffeomorphic to a real projective space.
