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

TL;DR

This paper investigates the limits of Riemannian 4-manifolds with specific curvature conditions related to their twistor spaces, showing that certain bubble formations are impossible in these limits.

Contribution

It establishes a curvature inequality for manifolds with taming forms on their twistor spaces and proves that hyperkähler limits cannot contain holomorphic 2-spheres, ruling out certain bubble formations.

Findings

Curvature inequality involving self-dual Weyl and Ricci curvature

Hyperkähler limits cannot contain holomorphic 2-spheres

Prevents formation of ALE gravitational instanton bubbles

Abstract

The twistor space of a Riemannian 4-manifold carries two almost complex structures, $J_+$ and $J_-$, and a natural closed 2-form $\omega$. This article studies limits of manifolds for which $\omega$ 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 hyperk\"ahler limit X (in the $C^2$ pointed topology) then X cannot contain a holomorphic 2-sphere (for any of its hyperk\"ahler complex structures). In particular, this rules out the formation of bubbles modelled on ALE gravitational instantons in such families of metrics.