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Contact Spheres and Hyperkähler Geometry

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Abstract

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.

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Authors and Affiliations

  1. Mathematisches Institut, Universität zu Köln, Weyertal 86–90, 50931, Köln, Germany

    Hansjörg Geiges

  2. Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049, Madrid, Spain

    Jesús Gonzalo Pérez

Authors
  1. Hansjörg Geiges
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  2. Jesús Gonzalo Pérez
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Correspondence to Hansjörg Geiges.

Additional information

Communicated by G. W. Gibbons

Partially supported by DFG grant GE 1245/1-2 within the framework of the Schwerpunktprogramm “Globale Differentialgeometrie”.

Partially supported by grants MTM2004-04794 and MTM2007-61982 from MEC Spain.

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Geiges, H., Pérez, J.G. Contact Spheres and Hyperkähler Geometry. Commun. Math. Phys. 287, 719–748 (2009). https://doi.org/10.1007/s00220-008-0634-y

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