• DOI:10.1007/s00220-008-0634-y
• Corpus ID: 17112312

Contact Spheres and Hyperkähler Geometry

@article{Geiges2001ContactSA,
  title={Contact Spheres and Hyperk{\"a}hler Geometry},
  author={Hansjorg Geiges and Jes'us Gonzalo P'erez},
  journal={Communications in Mathematical Physics},
  year={2001},
  volume={287},
  pages={719-748}
}

• H. Geiges, J. P'erez
• Published 10 October 2001
• Mathematics
• Communications in Mathematical Physics

A taut contact sphere on a 3-manifold is a linear 2-sphere of contact forms, all defining the same volume form. In the present paper we completely determine the moduli of taut contact spheres on compact left-quotients of SU(2) (the only closed manifolds admitting such structures). We also show that the moduli space of taut contact spheres embeds into the moduli space of taut contact circles.This moduli problem leads to a new viewpoint on the Gibbons-Hawking ansatz in hyperkähler geometry. The classification of taut contact spheres on closed 3-manifolds includes the known classification of 3-Sasakian 3-manifolds, but the local Riemannian geometry of contact spheres is much richer. We construct two examples of taut contact spheres on open subsets of $${\mathbb{R}^3}$$ with nontrivial local geometry; one from the Helmholtz equation on the 2-sphere, and one from the Gibbons-Hawking ansatz. We address the Bernstein problem whether such examples can give rise to complete metrics.