An ansatz for almost-Kähler, Einstein 4-manifolds (Q2769331)

This paper is motivated by Goldberg's conjecture saying that every compact, almost Kähler, Einstein manifold is necessarily Kähler. An almost Kähler manifold is an almost Hermitian manifold whose fundamental \(2\)-form \(\omega\) is closed. An almost Kähler manifold that is not Kähler is said to be strictly almost Kähler. The \(\ast\)-Ricci tensor \(R\omega\) on an almost Kähler manifold \(M\) is obtained by the usual action of the curvature \(R \in \text{End}(\Lambda^2)\) on \(\omega\), and if \(R\omega\) is a multiple of \(\omega\) then \(M\) is said to be weakly \(\ast\)-Einstein. The main result of the paper is: Any strictly almost Kähler, Einstein, weakly \(\ast\)-Einstein \(4\)-manifold is necessarily given by Tod's ansatz.NEWLINENEWLINENEWLINETod's ansatz is a method for generating almost Kähler Einstein metrics. It generalizes the Gibbons-Hawking ansatz, which describes a correspondence between hyper-Kähler manifolds and harmonic forms, and which was used by \textit{P. Nurowski} and \textit{M. Przanowski} [Classical Quantum Gravity 16, L9--L13 (1999; Zbl 0979.53045)] for constructing the first examples of strictly almost Kähler Einstein manifolds.NEWLINENEWLINENEWLINEA consequence of the main result is that these particular \(4\)-manifolds can never be compact and are always self-dual. The author's strategy for the proof is to reduce the problem to the Frobenius theorem rather than applying Cartan-Kähler theory. In combination with an investigation of the geometry of Tod's examples this implies the main result. The author also gives examples of how to use this approach in higher dimensions.