The J-flow on Kahler surfaces: a boundary case

TL;DR

This paper investigates the behavior of the J-flow on Kahler surfaces at the boundary of the convergence cone, revealing convergence to singular metrics and implications for the Mabuchi energy functional.

Contribution

It provides the first analysis of the J-flow at the boundary case, establishing convergence to singular metrics and exploring applications to energy functionals.

Findings

J-flow converges to a singular Kahler metric away from certain curves.

Established a C^0 estimate for the flow in the boundary case.

Discussed implications for the Mabuchi energy on Kahler surfaces.

Abstract

We study the J-flow on Kahler surfaces when the Kahler class lies on the boundary of the open cone for which global smooth convergence holds, and satisfies a nonnegativity condition. We obtain a C^0 estimate and show that the J-flow converges smoothly to a singular Kahler metric away from a finite number of curves of negative self-intersection on the surface. We discuss an application to the Mabuchi energy functional on Kahler surfaces with ample canonical bundle.