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.

In this paper, we prove a C 1,1 estimate for solutions of complex Monge-Ampère equations on compact almost Hermitian manifolds. Using this C 1,1 estimate, we show existence of C 1,1 solutions to the degenerate Monge-Ampère equations, the corresponding Dirichlet problems and the singular Monge-Ampère equations. We also study the singularities of the pluricomplex Green's function. In addition, the proof of the above C 1,1 estimate is valid for a kind of complex Monge-Ampère type equations. As a geometric application, we prove the C 1,1 regularity of geodesics in the space of Sasakian metrics.

We establish a maximum principle for a two-point function in order to analyze the convexity of level sets of harmonic functions. We show that this can be used to prove a strict convexity result involving the smallest principal curvature of the level sets.

We extend the continuity equation of La Nave-Tian to Hermitian metrics and establish its interval of maximal existence. The equation is closely related to the Chern-Ricci flow, and we illustrate this in the case of elliptic bundles over a curve of genus at least two.