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