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On positivity in Sasaki geometry

  • Charles P. BoyerEmail author
  • Christina W. Tønnesen-Friedman
Original Paper
  • 14 Downloads

Abstract

It is well known that if the dimension of the Sasaki cone \({\mathfrak {t}}^+\) is greater than one, then all Sasakian structures in \({\mathfrak {t}}^+\) are either positive or indefinite. We discuss the phenomenon of type changing within a fixed Sasaki cone. Assuming henceforth that \(\dim {\mathfrak {t}}^+>1\) there are three possibilities, either all elements of \({\mathfrak {t}}^+\) are positive, all are indefinite, or both positive and indefinite Sasakian structures occur in \({\mathfrak {t}}^+\). We illustrate by examples how the type can change as we move in \({\mathfrak {t}}^+\). If there exists a Sasakian structure in \({\mathfrak {t}}^+\) whose total transverse scalar curvature is non-positive, then all elements of \({\mathfrak {t}}^+\) are indefinite. Furthermore, we prove that if the first Chern class is a torsion class or represented by a positive definite (1, 1) form, then all elements of \({\mathfrak {t}}^+\) are positive.

Keywords

Positive Sasakian structures Positivity of Ricci curvature Join construction 

Mathematics Subject Classification (2000)

Primary: 53C25 

Notes

Acknowledgements

The authors are grateful to Vestislav Apostolov for carefully reading our paper and suggesting some important clarifications. We also thank Hongnian Huang, and Eveline Legendre for their interest in and comments on our work.

References

  1. 1.
    Apostolov, V., Calderbank, D.M.J., Gauduchon, P.: Hamiltonian 2-forms in Kähler geometry. I. General theory. J. Diff. Geom. 73(3), 359–412 (2006)CrossRefzbMATHGoogle Scholar
  2. 2.
    Apostolov, V., Calderbank, D.M.J., Gauduchon, P., Tønnesen-Friedman, C.W.: Hamiltonian 2-forms in Kähler geometry. III. Extremal metrics and stability. Invent. Math. 173(3), 547–601 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Apostolov, V., Calderbank, D.M.J., Gauduchon, P., Tønnesen-Friedman, C.W.: Hamiltonian 2-forms in Kähler geometry. IV. Weakly Bochner-flat Kähler manifolds. Commun. Anal. Geom. 16(1), 91–126 (2008)CrossRefzbMATHGoogle Scholar
  4. 4.
    Boyer, C.P., Galicki, K.: Highly connected manifolds with positive Ricci curvature. Geom. Topol. 10, 2219–2235 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Boyer, C.P., Galicki, K.: Sasakian Geometry. Oxford Mathematical Monographs. Oxford University Press, Oxford (2008)zbMATHGoogle Scholar
  6. 6.
    Boyer, C.P., Galicki, K., Nakamaye, M.: On positive Sasakian geometry. Geom. Dedicata 101, 93–102 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Boyer, C.P., Galicki, K., Nakamaye, M.: Sasakian geometry, homotopy spheres and positive Ricci curvature. Topology 42(5), 981–1002 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Boyer, C.P., Huang, H., Legendre, E.: An application of the Duistermaat-Heckman theorem and its extensions in Sasaki geometry. Geom. Topol. 22, 4205–4234 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Boyer, C.P., Huang, H., Legendre, E., Tønnesen-Friedman, C.W.: The Einstein–Hilbert functional and the Sasaki–Futaki invariant. Int. Math. Res. Not. IMRN 2017(7), 1942–1974 (2017)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Boyer, C.P.: Maximal tori in contactomorphism groups. Diff. Geom. Appl. 31(2), 190–216 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Boyer, C.P., Pati, J.: On the equivalence problem for toric contact structures on \(S^3\)-bundles over \(S^2\). Pac. J. Math. 267(2), 277–324 (2014)CrossRefzbMATHGoogle Scholar
  12. 12.
    Boyer, C.P., Tønnesen-Friedman, C.W.: Extremal Sasakian geometry on \(S^3\)-bundles over Riemann surfaces. Int. Math. Res. Not. IMRN 2014(20), 5510–5562 (2014)CrossRefzbMATHGoogle Scholar
  13. 13.
    Boyer, C.P., Tønnesen-Friedman, C.W.: Simply connected manifolds with infinitely many toric contact structures and constant scalar curvature Sasaki metrics (2014). Preprint arXiv:1404.3999
  14. 14.
    Boyer, C.P., Tønnesen-Friedman, C.W.: On the topology of some Sasaki–Einstein manifolds. NYJM 21, 57–72 (2015)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Boyer, C.P., Tønnesen-Friedman, C.W.: The Sasaki join and admissible Kähler constructions. J. Geom. Phys. 91, 29–39 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Boyer, C.P., Tønnesen-Friedman, C.W.: The Sasaki join, Hamiltonian 2-forms, and constant scalar curvature. J. Geom. Anal. 26, 1023–1060 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Calabi, E.: Extremal Kähler Metrics. Seminar on Differential Geometry, Ann. of Math. Stud., vol. 102, pp. 259–290. Princeton Univ. Press, Princeton (1982)zbMATHGoogle Scholar
  18. 18.
    El Kacimi-Alaoui, A.: Opérateurs transversalement elliptiques sur un feuilletage riemannien et applications. Compos. Math. 73(1), 57–106 (1990)zbMATHGoogle Scholar
  19. 19.
    Gauntlett, J.P., Martelli, D., Sparks, J., Waldram, D.: Sasaki–Einstein metrics on \(S^2\times S^3\). Adv. Theor. Math. Phys. 8(4), 711–734 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Guan, D.: Existence of extremal metrics on compact almost homogeneous Kähler manifolds with two ends. Trans. Am. Math. Soc. 347(6), 2255–2262 (1995)zbMATHGoogle Scholar
  21. 21.
    Hwang, A.D., Singer, M.A.: A momentum construction for circle-invariant Kähler metrics. Trans. Am. Math. Soc. 354(6), 2285–2325 (2002)CrossRefzbMATHGoogle Scholar
  22. 22.
    Hwang, A.D.: On existence of Kähler metrics with constant scalar curvature. Osaka J. Math. 31(3), 561–595 (1994)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Koiso, N.: On Rotationally Symmetric Hamilton’s Equation for Kähler–Einstein Metrics. Recent Topics in Differential and Analytic Geometry, Adv. Stud. Pure Math., vol. 18, pp. 327–337. Academic Press, Boston (1990)zbMATHGoogle Scholar
  24. 24.
    Koiso, N., Sakane, Y.: Nonhomogeneous Kähler–Einstein Metrics on Compact Complex Manifolds. Curvature and Topology of Riemannian Manifolds (Katata, 1985), Lecture Notes in Math., vol. 1201, pp. 165–179. Springer, Berlin (1986)zbMATHGoogle Scholar
  25. 25.
    LeBrun, C.: Scalar-flat Kähler metrics on blown-up ruled surfaces. J. Reine Angew. Math. 420, 161–177 (1991)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Martelli, D., Sparks, J., Yau, S.-T.: Sasaki–Einstein manifolds and volume minimisation. Commun. Math. Phys. 280(3), 611–673 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Nozawa, H.: Deformation of Sasakian metrics. Trans. Am. Math. Soc. 366(5), 2737–2771 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Pedersen, H., Poon, Y.S.: Hamiltonian constructions of Kähler–Einstein metrics and Kähler metrics of constant scalar curvature. Commun. Math. Phys. 136(2), 309–326 (1991)CrossRefzbMATHGoogle Scholar
  29. 29.
    Sakane, Y.: Examples of compact Einstein Kähler manifolds with positive Ricci tensor. Osaka J. Math. 23(3), 585–616 (1986)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Sullivan, D.: Infinitesimal computations in topology. Inst. Hautes Études Sci. Publ. Math. (1977) 47, 269–331 (1978)CrossRefzbMATHGoogle Scholar
  31. 31.
    Tønnesen-Friedman, C.W.: Extremal Kähler metrics on minimal ruled surfaces. J. Reine Angew. Math. 502, 175–197 (1998)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of New MexicoAlbuquerqueUSA
  2. 2.Department of MathematicsUnion CollegeSchenectadyUSA

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