  We study the sharp doubling inequalities for the gradients and upper bounds
for the critical sets of Dirichlet eigenfunctions on the boundary and in the
interior of compact Riemannian manifolds. Most efforts are devoted to obtaining
the sharp doubling inequalities for the gradients. New technique is developed
to overcome the difficulties on the unavailability of the double manifold in
obtaining doubling inequalities in smooth manifolds. The sharp upper bounds of
critical sets in analytic Riemannian manifolds are consequences of the doubling
inequalities.
