The calabi-yau equation on almost-kähler four-manifolds

Ben Weinkove*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

32 Scopus citations


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-Kähler 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 CalabiYau 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.

Original languageEnglish (US)
Pages (from-to)317-349
Number of pages33
JournalJournal of Differential Geometry
Issue number2
StatePublished - 2007

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Geometry and Topology

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