  We provide explicit examples which show that mean convexity (i.e. positivity
of the mean curvature) and positivity of the scalar curvature are non-preserved
curvature conditions for hypersurfaces of the Euclidean space evolving under
either the volume- or the area preserving mean curvature flow. The relevance of
our examples is that they disprove some statements of the previous literature,
overshadow a widespread folklore conjecture about the behaviour of these flows
and bring out the discouraging news that a traditional singularity analysis is
not possible for constrained versions of the mean curvature flow.
