On the convergence and singularities of the J-flow with applications to the Mabuchi energy

The J-flow of S. K. Donaldson and X. X. Chen is a parabolic flow on Kähler manifolds with two Kähler metrics. It is the gradient flow of the J -functional that appears in Chen's formula for the Mabuchi energy. We find a positivity condition in terms of the two metrics that is both necessary and sufficient for the convergence of the J-flow to a critical metric. We use this result to show that on manifolds with ample canonical bundle, the Mabuchi energy is proper on all Kähler classes in an open neighborhood of the canonical class defined by a positivity condition. This improves previous results of Chen and of the second author. We discuss the implications of this for the problem of the existence of constant-scalar-curvature Kähler metrics. We also study the singularities of the J-flow and, under certain conditions (which always hold for dimension 2) derive some estimates away from a subvariety. We discuss the conjectural remark of Donaldson that if the J-flow does not converge on a Kähler surface, then it should blow up over some curves of negative self-intersection.