Anti-self-dual 4-manifolds, quasi-Fuchsian groups, and almost-Kähler geometry (Q2208308)

An oriented Riemannian 4-manifold \((M^4, g)\) is said to be anti-self-dual if its self-dual Weyl curvature vanishes. For example, any oriented locally-conformally-flat 4-manifold is anti-self-dual. A Kähler manifold of complex dimension two is anti-self-dual if and only if it is scalar-flat. Among anti-self-dual conformal classes, those that can be represented by Kähler metrics are highly non-generic. The authors illustrates this fact using the example \(M=\Sigma\times \mathbb{C}P_1\) where \(\Sigma\) is closed Riemann surface of genus \(g\geq 2\). The moduli space of anti-self-dual conformal classes on \(M\) has dimension \(30(g-1)\), but only a subspace of dimension \(12(g-1)\) arises from conformal structures that contain scalar-flat Kähler metrics. One then considers a larger class of almost Kähler anti-self-dual metrics to provide a possible model for general anti-self-dual metrics. Almost Kähler anti-self-dual metrics share many of the remarkable properties of the scalar-flat Kähler metrics. More importantly, the almost Kähler condition is open in the moduli space of anti-self-dual conformal classes. This paper asks whether almost-Kähler condition is also closed in the anti-self-dual context and shows that the answer is no. The proof is by constructing a family of metrics on the 4-manifold \(M=\Sigma\times S^2\), where \(\Sigma\) is a closed oriented Riemann surface of even large genus. Indeed, the authors show that the 4-manifold \(M\) admits locally-conformally-flat conformal classes \([g]\) that cannot be represented by almost-Kähler metrics, but are deformations of scalar-flat Kähler metric on \(M\). Such \((M,[g])\) are anti-self-dual 4-manifold arising via the hyperbolic ansatz from a quasi-Fuchsian 3-manifold of Bers type. The examples described above are all locally-conformally-flat, and thus live on 4-manifolds of signature zero. A similar construction in this paper produces many explicit examples showing that the almost Kähler condition is not closed living on 4-manifolds with negative sigature, and so are not locally conformally flat.