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

By B. Weinkove. 2006

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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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/9c71400f338e6a1806c85460e4832f2dfff29562

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Regularity results for pluriclosed flow (2010, background): “Related quantities for Monge– Ampère-type equations on almost Kähler surfaces were considered in Weinkove [33].”
Complex Monge-Ampere equations and totally real submanifolds (2009, background): “In [24], Donaldson proposed to generalize Yau’s theorems in a different direction; see [69,68, 65,66] for some recent developments.”
$$C^{2,\alpha }$$C2,α estimates for nonlinear elliptic equations in complex and almost complex geometry (2014, methodology): “Some partial results towards this conjecture were given in [62,70].”
A correction to “Tamed symplectic forms and strong Kähler with torsion metrics” (2012, background): “Moreover, the study of tamed symplectic structures in dimension 4 is related to a more general conjecture of Donaldson (see for instance [7, 28, 44]).”
Taming symplectic forms and the Calabi–Yau equation (2007, methodology): “In fact, this section is not strictly necessary to complete the proofs of the theorems, since we could have instead appealed to the argument in [17], adapting the technique of Evans [6] and Krylov [12].”
The Monge–Ampère equation for non-integrable almost complex structures (2016, background): “Donaldson’s conjecture remains open in general, but it is known to hold in some special cases [55, 52, 47, 48, 16, 6].”