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Toric Self-Dual Einstein Metrics as Quotients

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We use the quaternion Kähler reduction technique to study old and new self-dual Einstein metrics of negative scalar curvature with at least a two-dimensional isometry group, and relate the quotient construction to the hyperbolic eigenfunction Ansatz. We focus in particular on the (semi-)quaternion Kähler quotients of (semi-)quaternion Kähler hyperboloids, analysing the completeness and topology, and relating them to the self-dual Einstein Hermitian metrics of Apostolov–Gauduchon and Bryant.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM, 87131, USA

    Charles P. Boyer & Krzysztof Galicki

  2. School of Mathematics, University of Edinburgh, King’s Buildings, Mayfield Road, Edinburgh, EH9 3JZ, Scotland

    David M.J. Calderbank

  3. Università degli Studi di Roma, “La Sapienza”, Dipartimento di Matematica, Piazzale Aldo Moro, 00185, Roma, Italia

    Paolo Piccinni

Authors
  1. Charles P. Boyer
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  2. David M.J. Calderbank
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  3. Krzysztof Galicki
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  4. Paolo Piccinni
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Corresponding author

Correspondence to Charles P. Boyer.

Additional information

Communicated by G.W. Gibbons

During the preparation of this work the first and third authors were supported by NSF grant DMS-0203219. The second author was supported by the Leverhulme Trust, the William Gordon Seggie Brown Trust and an EPSRC Advanced Fellowship. The fourth author was supported by the MIUR Project “Proprietà Geometriche delle Varietà Reali e Complesse”. The authors are also grateful for support from EDGE, Research Training Network HPRN-CT-2000-00101, funded by the European Human Potential Programme.

Acknowledgement The first author thanks the Ecole Polytechnique, Palaiseau and the Università di Roma “La Sapienza” for hospitality and support. The third author would like to thank the Università di Roma “La Sapienza”, I.N.d.A.M, M.P.I-Bonn, and IHES as parts of this paper were written during his visits there. The fourth named author would like to thank University of New Mexico for hospitality and support. The authors are grateful to Paul Gauduchon, Michael Singer and Pavel Winternitz for invaluable discussions.

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Boyer, C., Calderbank, D., Galicki, K. et al. Toric Self-Dual Einstein Metrics as Quotients. Commun. Math. Phys. 253, 337–370 (2005). https://doi.org/10.1007/s00220-004-1192-6

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