Communications in Mathematical Physics

, Volume 279, Issue 3, pp 705–733 | Cite as

Canonical Sasakian Metrics

  • Charles P. Boyer
  • Krzysztof Galicki
  • Santiago R. Simanca


Let M be a closed manifold of Sasaki type. A polarization of M is defined by a Reeb vector field, and for any such polarization, we consider the set of all Sasakian metrics compatible with it. On this space we study the functional given by the square of the L 2-norm of the scalar curvature. We prove that its critical points, or canonical representatives of the polarization, are Sasakian metrics that are transversally extremal. We define a Sasaki-Futaki invariant of the polarization, and show that it obstructs the existence of constant scalar curvature representatives. For a fixed CR structure of Sasaki type, we define the Sasaki cone of structures compatible with this underlying CR structure, and prove that the set of polarizations in it that admit a canonical representative is open. We use our results to describe fully the case of the sphere with its standard CR structure, showing that each element of its Sasaki cone can be represented by a canonical metric; we compute their Sasaki-Futaki invariant, and use it to describe the canonical metrics that have constant scalar curvature, and to prove that only the standard polarization can be represented by a Sasaki-Einstein metric.


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  1. 1.
    Boyer C.P. and Galicki K. (2000). A note on toric contact geometry. J. Geom. Phys. 35: 288–298 MATHCrossRefADSMathSciNetGoogle Scholar
  2. 2.
    Boyer C.P. and Galicki K. (2001). Einstein manifolds and contact geometry. Proc. Amer. Math. 129: 2419–2430 MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Boyer C.P. and Galicki K. (2008). Sasakian Geometry. Oxford Mathematical Monographs. Oxford University Press, Oxford MATHGoogle Scholar
  4. 4.
    Boyer C.P. and Galicki K. (2005). Sasakian geometry, hypersurface singularities and Einstein metrics. Rend. Circ. Mat. Palermo (2) Suppl. 75: 57–87 MathSciNetGoogle Scholar
  5. 5.
    Boyer C.P., Galicki K. and Kollár J. (2005). Einstein metrics on spheres. Ann. of Math. 162: 557–580 MATHMathSciNetGoogle Scholar
  6. 6.
    Boyer C.P., Galicki K. and Matzeu P. (2006). On Eta-Einstein Sasakian Geometry. Commun. Math. Phys. 262: 177–208 MATHCrossRefADSMathSciNetGoogle Scholar
  7. 7.
    Brinzanescu V. and Slobodeanu R. (2006). Holomorphicity and Walczak formula on Sasakian manifolds. J. Geom. and Phys. 57(1): 193–207 CrossRefADSMathSciNetGoogle Scholar
  8. 8.
    Bryant R.L. (2001). Bochner-Kähler metrics. J. Amer. Math. Soc. 14: 623–715 MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Calabi E. (1982). Extremal Kähler metrics. In: Yau, S.T. (eds) Seminar of Differerential Geometry, Annals of Math. Studies,102., pp 259–290. Princeton University Press, Princeton, NJ Google Scholar
  10. 10.
    Calabi, E.: Extremal Kähler metrics II. In: Differential geometry and complex analysis I. Chavel, H.M. Farkas, eds. Berlin-Heidelberg-New York: Springer-Verlag, 1985, pp. 95–114Google Scholar
  11. 11.
    Calabi, E.: The Space of Kähler Metrics. Proc. Int. Cong. Math., Amsterdam, Vol. 2, 1954, pp. 206–207Google Scholar
  12. 12.
    Cheeger J. and Tian G. (1994). On the cone structure at infinity of Ricci flat manifolds with Euclidean volume growth and quadratic curvature decay. Invent. Math. 118: 493–571 MATHCrossRefADSMathSciNetGoogle Scholar
  13. 13.
    Chern S.S. and Moser J.K. (1974). Real hypersurfaces in complex manifolds. Acta Math. 133: 219–271 CrossRefMathSciNetGoogle Scholar
  14. 14.
    Cvetic M., Lü H., Page D.N. and Pope C.N. (2005). New Einstein-Sasaki spaces in five and higher dimensions. Phys. Rev. Lett. 95: 4 CrossRefGoogle Scholar
  15. 15.
    David, L.: The Bochner-flat cone of a CR manifold., 2005
  16. 16.
    David, L., Gauduchon, P.: The Bochner-flat geometry of weighted projective spaces. C.R.M. Proceedings and Lecture Notes 40, Providence, RI: Amer. Math. Soc., 2006, pp. 104–156Google Scholar
  17. 17.
    Futaki A. (1983). An obstruction to the existence of Einstein Kähler metrics. Invent. Math. 73: 437–443 MATHCrossRefADSMathSciNetGoogle Scholar
  18. 18.
    Gauntlett J.P., Martelli D., Sparks J. and Waldram W. (2004). Sasaki-Einstein metrics on S 2 × S 3 . Adv. Theor. Math. Phys. 8: 711–734 MATHMathSciNetGoogle Scholar
  19. 19.
    Gauntlett J.P., Martelli D., Sparks J. and Waldram W. (2004). A new infinite class of Sasaki-Einstein manifolds. Adv. Theor. Math. Phys. 8: 987–1000 MATHMathSciNetGoogle Scholar
  20. 20.
    Kollár J. (2006). Circle actions on simply connected 5-manifolds. Topology 45: 643–671 MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Kollár J. (2005). Einstein metrics on five-dimensional Seifert bundles. J. Geom. Anal. 15: 445–476 MATHMathSciNetGoogle Scholar
  22. 22.
    LeBrun, C., Simanca, S.R.: On the Kähler Classes of Extremal Metrics. In: Geometry and Global Analysis, (First MSJ Intern. Res. Inst. Sendai, Japan) eds. T. Kotake, S. Nishikawa, R. Schoen, Sendai. Tohoku Univ. Press, 1993, pp. 255–271Google Scholar
  23. 23.
    Lee J.M. (1996). CR manifolds with noncompact connected automorphism groups. J. Geom. Anal. 6: 79–90 MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Lichnerowicz A. (1957). Sur les transformations analytiques des variétés kählériennes compactes. C.R. Acad. Sci. Paris 244: 3011–3013 MATHMathSciNetGoogle Scholar
  25. 25.
    Martelli D. and Sparks J. (2005). Toric Sasaki-Einstein metrics on S 2 × S 3 . Phys. Lett. B 621: 208–212 CrossRefADSMathSciNetGoogle Scholar
  26. 26.
    Martelli D. and Sparks J. (2006). Toric geometry, Sasaki-Einstein manifolds and a new infinite class of AdS/CFT duals. Commun. Math. Phys. 262: 51–89 MATHCrossRefADSMathSciNetGoogle Scholar
  27. 27.
    Martelli D., Sparks J. and Yau S.-T. (2005). The geometric dual of a-maximisation for toric Sasaki-Einstein manifolds. Commun. Math. Phys. 268: 39–65 CrossRefADSMathSciNetGoogle Scholar
  28. 28.
    Martelli, D., Sparks, J., Yau, S.-T.: Sasaki-Einstein manifolds and volume minimisation., 2006
  29. 29.
    Matsushima Y. (1957). Sur la structure du groupe d’homéomorphismes analytiques d’une certaine variété kählérienne. Nagoya Math. J. 11: 145–150 MATHMathSciNetGoogle Scholar
  30. 30.
    Molino, P.: Riemannian Foliations. Progress in Mathematics 73, Boston, MA: Birkhäuser Boston Inc., 1988Google Scholar
  31. 31.
    Nishikawa S. and Tondeur P. (1988). Transversal infinitesimal automorphisms for harmonic Kähler foliations. Tôhuku Math. J. 40: 599–611 MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    O’Neill B. (1966). The fundamental equations of a submersion. Mich. Math. J. 13: 459–469 MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Schoen R. (1995). On the conformal and CR automorphism groups. Geom. Funct. Anal. 5: 464–481 MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Simanca, S.R.: Canonical Metrics on Compact Almost Complex Manifolds. Publicações Matemáticas do IMPA, Rio de Janeiro: IMPA, 2004, 97 pp.Google Scholar
  35. 35.
    Simanca S.R. (2005). Heat Flows for Extremal Kähler Metrics. Ann. Scuola Norm. Sup. Pisa CL. Sci. 4: 187–217 MATHMathSciNetGoogle Scholar
  36. 36.
    Webster S.M. (1977). On the transformation group of a real hypersurface. Trans. Amer. Math. Soc. 231: 179–190 MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Charles P. Boyer
    • 1
  • Krzysztof Galicki
    • 1
  • Santiago R. Simanca
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of New MexicoAlbuquerqueUSA

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