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A Construction of Einstein-Weyl Spaces via LeBrun-Mason Type Twistor Correspondence

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Abstract

We construct infinitely many Einstein-Weyl structures on \({S^2\times\mathbb{R}}\) of signature (− + +) which is sufficiently close to the model case of constant curvature, and on which the space-like geodesics are all closed. Such a structure is obtained as a parameter space of a family of holomorphic disks which is associated to a small perturbation of the diagonal of \({\mathbb{CP}^1\times\overline{\mathbb{CP}^1}}\) . The geometry of constructed Einstein-Weyl spaces is well understood from the configuration of holomorphic disks. We also review Einstein-Weyl structures and their properties in the former half of this article.

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References

  1. Calderbank, D.M.J.: Selfdual 4-manifolds, projective surfaces, and the Dunajski-West construction. http://arxiv.org/abs/math.DG/0606754, 2006

  2. Dunajski M.: A class of Einstein-Weyl spaces associated to an integrable system of hydrodynamic type. J. Geom. Phys. 51(1), 126–137 (2004)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  3. Dunajski M., Mason L.J., Tod P.: Einstein-Weyl geometry, the dKP equation and twistor theory. J. Geom. Phys. 37(1–2), 63–93 (2001)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  4. Dunajski M., West S.: Anti-self-dual conformal structures with null Killing vectors from projective structures. Commum. Math. Phys. 272(1), 85–118 (2007)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  5. Dunajski, M., West, S.: Anti-self-dual conformal structures in neutral signature. http://arxiv.org/abs/math/0610280V4[math.DG], 2008, to appear in Recent Developments in pseudo In: Riemannian geometry, ESI-Series on Math and Physics

  6. Hitchin, N.J.: Complex manifolds and Einstein’s equations. In: Twistor Geometry and Non-Linear Systems, Lecture Notes in Mathematics, Vol. 970, 1982

  7. Jones P.E., Tod K.P.: Minitwistor spaces and Einstein-Weyl spaces. Class. Quant. Grav. 2, 565–577 (1985)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  8. LeBrun, C.: Twistors, Holomorphic Disks, and Riemann Surfaces with Boundary. In: Perspectives in Riemannian geometry, CRM Proc. Lecture Notes, 40, Providence, RI: Amer. Math. Soc. 2006, pp. 209–221

  9. LeBrun C., Mason L.J.: Zoll Manifolds and complex surfaces. J. Diff. Geom. 61, 453–535 (2002)

    MATH  MathSciNet  Google Scholar 

  10. LeBrun C., Mason L.J.: Nonlinear Gravitons, Null Geodesics, and Holomorphic Disks. Duke Math. J. 136(2), 205–273 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  11. Nakata F.: Singular self-dual Zollfrei metrics and twistor correspondence. J. Geom. Phys. 57(6), 1477–1498 (2007)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  12. Nakata F.: Self-dual Zollfrei conformal structures with α-surface foliation. J. Geom. Phys. 57(10), 2077–2097 (2007)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  13. Pedersen H.: Einstein-Weyl spaces and (1,n)-curves in the quadric surface. Ann. Global Anal. Geom. 4(1), 89–120 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  14. Pedersen H., Tod K.P.: Three-dimensional Einstein-Weyl geometry. Adv. Math. 97, 74–109 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  15. Penrose R.: Nonlinear gravitons and curved twistor theory. Gen. Rel. Grav. 7, 31–52 (1976)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  16. Tod K.P.: Compact 3-dimensional Einstein-Weyl structures. J. London Math. Soc (2) 45, 341–351 (1992)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Graduate School of Science and Engineering, Tokyo Institute of Technology, 2-12-1, O-okayama, Meguro, 152-8551, Japan

    Fuminori Nakata

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  1. Fuminori Nakata
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Correspondence to Fuminori Nakata.

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Communicated by G.W. Gibbons

This work is partially supported by Grant-in-Aid for Scientific Research of the Japan Society for the Promotion of Science.

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Nakata, F. A Construction of Einstein-Weyl Spaces via LeBrun-Mason Type Twistor Correspondence. Commun. Math. Phys. 289, 663–699 (2009). https://doi.org/10.1007/s00220-009-0750-3

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