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

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 C0 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.

Kähler, $J$-flow, complex Monge–Ampère
Mathematical Subject Classification 2010
Primary: 53C44, 53C55
Received: 9 January 2013
Accepted: 22 August 2013
Published: 7 May 2014
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