#### Vol. 7, No. 1, 2014

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The $J$-flow on Kähler surfaces: a boundary case

### Hao Fang, Mijia Lai, Jian Song and Ben Weinkove

Vol. 7 (2014), No. 1, 215–226
##### Abstract

We study the $J$-flow on Kähler surfaces when the Kähler 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 Kähler 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 Kähler surfaces with ample canonical bundle.

##### Keywords
Kähler, $J$-flow, complex Monge–Ampère
##### Mathematical Subject Classification 2010
Primary: 53C44, 53C55
##### Milestones
Accepted: 22 August 2013
Published: 7 May 2014
##### Authors
 Hao Fang Department of Mathematics University of Iowa Iowa City, IA 52242 United States Mijia Lai Department of Mathematics University of Rochester Rochester, NY 14642 United States Jian Song Department of Mathematics Rutgers University Piscataway, NJ 08854 United States Ben Weinkove Mathematics Department Northwestern University Evanston, IL 60208 United States