  This is the second of a series of two papers where we construct embedded
Willmore tori with small area constraint in Riemannian three-manifolds. In both
papers the construction relies on a Lyapunov-Schmidt reduction, the difficulty
being the M\"obius degeneration of the tori. In the first paper the
construction was performed via minimization, here by Morse Theory; to this aim
we establish new geometric expansions of the derivative of the Willmore
functional on exponentiated small Clifford tori degenerating, under the action
of the M\"obius group, to small geodesic spheres with a small handle. By using
these sharp asymptotics we give sufficient conditions, in terms of the ambient
curvature tensors and Morse inequalities, for having existence/multiplicity of
embedded tori stationary for the Willmore functional under the constraint of
prescribed (sufficiently small) area.
