  The J-flow is a parabolic flow on Kahler manifolds. It was defined by
Donaldson in the setting of moment maps and by Chen as the gradient flow of the
J-functional appearing in his formula for the Mabuchi energy. It is shown here
that under a certain condition on the initial data, the J-flow converges to a
critical metric. This is a generalization to higher dimensions of the author's
previous work on Kahler surfaces. A corollary of this is the lower boundedness
of the Mabuchi energy on Kahler classes satisfying a certain inequality when
the first Chern class of the manifold is negative.
