Convergence of the J-flow on Kähler surfaces

Donaldson defined a parabolic flow of potentials on Kähler manifolds which arises from considering the action of a group of symplectomorphisms on the space of smooth maps between manifolds. One can define a moment map for this action, and then consider the gradient flow of the square of its norm. Chen discovered the same flow from a different viewpoint and called it the J-flow, since it corresponds to the gradient flow of his J-functional, which is related to Mabuchi's K-energy. In this paper, we show that in the case of Kähler surfaces with two Kähler forms satisfying a certain inequality, the J-flow converges to a zero of the moment map.