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The Calabi-Yau equation on almost-Kähler four-manifolds

Ben Weinkove Harvard University

Differential Geometry mathscidoc:1609.10024

Journal of Differential Geometry, 76, (2), 317-349, 2007
Let (M, ω) be a compact symplectic 4-manifold with a compatible almost complex structure J. The problem of finding a J-compatible symplectic form with prescribed volume form is an almost-K¨ahler analogue of Yau’s theorem and is connected to a programme in symplectic topology proposed by Donaldson. We call the corresponding equation for the symplectic form the Calabi- Yau equation. Solutions are unique in their cohomology class. It is shown in this paper that a solution to this equation exists if the Nijenhuis tensor is small in a certain sense. Without this assumption, it is shown that the problem of existence can be reduced to obtaining a C0 bound on a scalar potential function.
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@inproceedings{ben2007the,
  title={The Calabi-Yau equation on almost-Kähler four-manifolds},
  author={Ben Weinkove},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160908201507098928681},
  booktitle={Journal of Differential Geometry},
  volume={76},
  number={2},
  pages={317-349},
  year={2007},
}
Ben Weinkove. The Calabi-Yau equation on almost-Kähler four-manifolds. 2007. Vol. 76. In Journal of Differential Geometry. pp.317-349. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160908201507098928681.
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