Limits of Riemannian 4‐manifolds and the symplectic geometry of their twistor spaces

The twistor space of a Riemannian 4‐manifold carries two almost complex structures, J+ and J− , and a natural closed 2‐form ω . This article studies limits of manifolds for which ω tames either J+ or J− . This amounts to a curvature inequality involving self‐dual Weyl curvature and Ricci curvature, and which is satisfied, for example, by all anti‐self‐dual Einstein manifolds with non‐zero scalar curvature. We prove that if a sequence of manifolds satisfying the curvature inequality converges to a hyperkähler limit X (in the C2 pointed topology), then X cannot contain a holomorphic 2‐sphere (for any of its hyperkähler complex structures). In particular, this rules out the formation of bubbles modelled on asymptotically locally Euclidean gravitational instantons in such families of metrics.