Skip to main content
SpringerLink
Log in

Extremal K-contact metrics

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

Extending a result of He to the non-integrable case of K-contact manifolds, it is shown that the transverse Hermitian scalar curvature may be interpreted as a moment map for the strict contactomorphism group. As a consequence, we may generalize the Sasaki–Futaki invariant to K-contact geometry and establish a number of elementary properties. Moreover, we prove that in dimension 5 certain deformation-theoretic results can be established also under weaker integrability conditions by exploiting the relationship between J-anti-invariant and self-dual 2-forms.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Apostolov, V., Calderbank, D.M.J., Gauduchon, P., Tønnesen-Friedman, C.W.: Extremal Kähler metrics on projective bundles over a curve. Adv. Math. 227(6), 2385–2424 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  2. Apostolov, V., Drǎghici, T.: The curvature and the integrability of almost-Kähler manifolds: a survey. Fields Inst. Commun. Ser. 35, 25–53 (2003)

    Google Scholar 

  3. Boyer, C.P.: Maximal tori in contactomorphism groups. Differ. Geom. Appl. 31(2), 190–216 (2013)

    Article  MATH  Google Scholar 

  4. Boyer, C.P., Galicki, K.: Sasaki Geometry. Oxford Mathematical Monographs. Oxford University Press, Oxford (2008)

    Google Scholar 

  5. Boyer, C.P., Galicki, K., Simanca, S.R.: Canonical Sasaki metrics. Commun. Math. Phys. 279(3), 705–733 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Boyer, C.P., Galicki, K., Simanca, S.R.: The Sasaki cone and extremal Sasaki metrics, Riemannian topology and geometric structures on manifolds. Progr. Math. 271, 263–290 (2009)

    MathSciNet  Google Scholar 

  7. Calabi, E.: Extremal Kähler metrics, in Seminar of Differential Geometry. In: Yau, S.T. (ed.) Annals of Mathematics Studies, vol. 102, pp. 259–290. Princeton University Press, Princeton (1982)

    Google Scholar 

  8. Calabi, E.: Extremal Kähler metrics II, Differential geometry and complex analysis, 95–114. Springer, Berlin (1985)

    Google Scholar 

  9. Drǎghici, T., Li, T.-J., Zhang, W.: On the J-anti-invariant cohomology of almost complex 4-manifolds. Q. J. Math. 64(1), 83–111 (2013)

    Article  MathSciNet  Google Scholar 

  10. Donaldson, S.K.: Remarks on Gauge theory, complex geometry and 4-manifold topology. In: Atiyah, M., Iagolnitzer, D. (eds.) The Fields Medallists Lectures, pp. 384–403. World Scientific, Singapore (1997)

    Chapter  Google Scholar 

  11. Fujiki, A.: Moduli space of polarized algebraic manifolds and Kähler metrics. Sugaku Expos. 5, 173–191 (1992)

    MathSciNet  Google Scholar 

  12. Fujiki, A., Schumacher, G.: The moduli space of extremal compact Kähler manifolds and generalized Weil–Petersson metrics. Publ. Res. Inst. Math. Sci. 26(1), 101–183 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  13. Futaki, A.: An obstruction to the existence of Einstein Kähler metrics. Invent. Math. 73, 437–443 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  14. Futaki, A., Mabushi, T.: Bilinear forms and extremal Kähler vector fields associated with Kähler classes. Math. Ann. 301, 199–210 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  15. Futaki, A., Ono, H., Wang, G.: Transverse Kähler geometry of Sasaki manifolds and toric Sasaki–Einstein manifolds. J. Diif. Geom. 83, 585–636 (2009)

  16. Gauduchon, P.: Calabi’s extremal Kähler metrics: an elementary introduction (in preparation)

  17. Gauduchon, P.: Hermitian connections and Dirac operators. Boll. della Unione Mat. Ital. B 11(Suppl 2), 257–288 (1997)

    MATH  MathSciNet  Google Scholar 

  18. Geiges, H.: An Introduction to Contact Topology. Cambridge Studies in Advanced Mathematics, vol 109. Cambridge University Press, Cambridge (2008)

    Book  Google Scholar 

  19. Gauntlett, J.P., Martelli, D., Sparks, J., Waldram, W.: Sasaki–Einstein metrics on \(S^2\times S^3\). Adv. Theor. Math. Phys. 8, 711–734 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  20. Gray, J.W.: Some global properties of contact structures. Ann. Math. 2(69), 421–450 (1959)

    Article  Google Scholar 

  21. He, H.: On the transverse scalar curvature of a compact Sasaki manifold (2011). arXiv:1105.4000

  22. El Kacimi-Alaoui, A.: Opérateurs transversalement elliptiques sur un feuilletage riemannien et applications. Compos. Math. 79, 57–106 (1990)

    Google Scholar 

  23. El. Kacimi-Alaoui, A., Gmira, B.: Stabilité du caractère kählérien transverse. Israel J. Math. 101, 323–347 (1997)

    Article  MathSciNet  Google Scholar 

  24. Kodaira, K., Morrow, J.: Complex Manifolds. Holt, Rinehart and Winston (1971)

    MATH  Google Scholar 

  25. LeBrun, C., Simanca, S.R.: On the Kähler classes of extremal metrics, Geometry and Global Analysis (Sendai, Japan 1993), FirstMath. Soc. Japan Intern. Res. Inst. Eds. Kotake, Nishikawa and Schoen

  26. Legendre, E.: Existence and non-uniqueness of constant scalar curvature toric Sasaki metrics. Compos. Math. 147(5), 1613–1634 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  27. Lejmi, M.: Extremal almost-Kähler metrics. Int. J. Math. 21(12), 1639–1662 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  28. Lejmi, M.: Stability under deformations of extremal almost-Kähler metrics in dimension \(4\). Math. Res. Lett. 17(4), 601–612 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  29. Lejmi, M.: Stability under deformations of Hermite-Einstein almost-Kähler metrics. Annales de l’institut Fourier 64(6), 22512263 (2014)

    Article  MathSciNet  Google Scholar 

  30. Libermann, P.: Sur les connexions hermitiennes. C. R. Acad. Sci. Paris 239, 1579–1581 (1954)

    MATH  MathSciNet  Google Scholar 

  31. Simanca, S.R.: Canonical metrics on compact almost complex manifolds, Publicações Matemáticas do IMPA, IMPA, Rio de Janeiro, 97 pp (2004)

  32. Simanca, S.R.: Heat flows for extremal Kähler metrics. Ann. Scuola Norm. Sup. Pisa CL. Sci. 4, 187–217 (2005)

    MATH  MathSciNet  Google Scholar 

  33. Sparks, J.: Sasakian–Einstein Manifolds. Surv. Diff. Geom. 16, 265–324 (2011)

  34. Tondeur, P.: Geometry of Riemannian foliations, volume 20 of Seminar on Mathematical Sciences. Keio University, Department of Mathematics, Yokohama (1994)

Download references

Acknowledgments

The first named author is very grateful to Christina Tønnesen-Friedman and Charles Boyer for useful discussions. The first named author is also grateful to Vestislav Apostolov for helpful comments. Both authors are thankful to Joel Fine and Weiyong He for several useful discussions.

Author information

Authors and Affiliations

  1. Département de Mathématiques, Université Libre de Bruxelles, CP218, Boulevard du Triomphe, 1050, Brussels, Belgium

    Mehdi Lejmi

  2. Institut für Mathematik, Universität Augsburg, Universitätsstrasse 14, 86159, Augsburg, Germany

    Markus Upmeier

Authors
  1. Mehdi Lejmi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Markus Upmeier
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mehdi Lejmi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lejmi, M., Upmeier, M. Extremal K-contact metrics. Math. Z. 281, 673–687 (2015). https://doi.org/10.1007/s00209-015-1503-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-015-1503-y

Keywords

Mathematics Subject Classification

Search

Navigation