Limits of Riemannian 4-manifolds and the symplectic geometry of their twistor spaces
Abstrak
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 $C^2$ 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 ALE gravitational instantons in such families of metrics.
Penulis (1)
Joel Fine
Akses Cepat
- Tahun Terbit
- 2016
- Bahasa
- en
- Sumber Database
- arXiv
- Akses
- Open Access ✓