  We consider the mean curvature flow of a closed hypersurface in the complex
or quaternionic projective space. Under a suitable pinching assumption on the
initial data, we prove apriori estimates on the principal curvatures which
imply that the asymptotic profile near a singularity is either strictly convex
or cylindrical. This result generalizes to a large class of symmetric ambient
spaces the estimates obtained in the previous works on the mean curvature flow
of hypersurfaces in Euclidean space and in the sphere.
