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

Details The Calabi-Yau equation on almost-Kahler four-manifolds The Calabi-Yau equation on almost-Kahler four-manifolds

2006-04-18

arxiv.org/abs/math/0604408v2

Mathematics

Differential Geometry

33 pages. Final version. Minor modifications. To appear in J. Differential Geom

Scientific paper

Let (M, \omega) 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 C^0 bound on a scalar potential function.

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