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

TL;DR

This paper establishes a necessary and sufficient positivity condition for the convergence of the J-flow on Kahler manifolds, explores its implications for the Mabuchi energy and constant scalar curvature metrics, and analyzes singularities especially in dimension two.

Contribution

It provides a new criterion for J-flow convergence, improves understanding of Mabuchi energy properness, and investigates singularity behavior in Kahler surfaces.

Findings

Positivity condition characterizes J-flow convergence.

Mabuchi energy is proper near the canonical class on manifolds with ample canonical bundle.

In dimension two, singularity estimates support Donaldson's conjecture on blow-up behavior.

Abstract

The J-flow of S. K. Donaldson and X. X. Chen is a parabolic flow on Kahler manifolds with two Kahler metrics. It is the gradient flow of the J-functional which appears in Chen's formula for the Mabuchi energy. We find a positivity condition in terms of the two metrics which 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 Kahler 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 Kahler metrics. We also study the singularities of the J-flow and, under certain conditions (which always hold for dimension two) derive estimates away from a subvariety. In the case of Kahler surfaces we use these estimates to confirm, at least in a certain sense, a conjectural remark of Donaldson that if the J-flow does not converge then it should blow up over some curves of negative self-intersection.