Skip to main content
SpringerLink
Log in

A gap theorem for positive Einstein metrics on the four-sphere

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

We show that there exists a universal positive constant \(\varepsilon _0 > 0\) with the following property: let g be a positive Einstein metric on the four-sphere \(S^4\). If the Yamabe constant of the conformal class [g] satisfies

$$\begin{aligned} Y(S^4, [g]) >\frac{1}{\sqrt{3}} Y(S^4, [g_{\mathbb S}]) - \varepsilon _0\,, \end{aligned}$$

where \(g_{\mathbb S}\) denotes the standard round metric on \(S^4\), then, up to rescaling, g is isometric to \(g_{\mathbb S}\). This is an extension of Gursky’s gap theorem for positive Einstein metrics on \(S^4\).

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. Akutagawa, K., Botvinnik, B., Kobayashi, O., Seshadri, H.: The Weyl functional near the Yamabe invariant. J. Geom. Anal. 13, 1–20 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Anderson, M.: Ricci curvature bounds and Einstein metrics on compact manifolds. J. Am. Math. Soc. 2, 455–490 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  3. Anderson, M.: Einstein metrics with prescribed conformal infinity on \(4\)-manifolds. Geom. Funct. Anal. 18, 305–366 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bando, S.: Bubbling out of Einstein manifolds. Tohoku Math. 42, 205–216 (1990). [Correction and addition. Tohoku Math. J. 42, 587–588 (1990)]

    Article  MathSciNet  MATH  Google Scholar 

  5. Bando, S., Kasue, A., Nakajima, H.: On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth. Invent. Math. 97, 313–349 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bartnik, R.: The mass of an asymptotically flat manifold. Comm. Pure Appl. Math. 39, 661–693 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  7. Besse, A.: Einstein Manifolds. Springer, Berlin (1987)

    Book  MATH  Google Scholar 

  8. Besson, G., Courtois, G., Gallot, S.: Entropies et rigidités des espaces localement symétriques de courbure strictement négative. Geom. Funct. Anal. 5, 731–799 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  9. Böhm, C.: Inhomogeneous Einstein metrics on low-dimensional spheres and other low-dimensional spaces. Invent. Math. 134, 145–176 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bourguignon, J.-P., Karcher, H.: Curvature operators: pinching estimates and geometric examples. Ann. Sci. École Norm. Sup. 11, 71–92 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  11. Crisp, J.S., Hillman, J.A.: Embedding Seifert fibred and \(3\)-manifolds and \({{\rm Sol}}^{3}\)-manifolds in \(4\)-space. Proc. Lond. Math. Soc. 76, 685–710 (1998)

    Article  MATH  Google Scholar 

  12. Ebin, D.: The manifold of Riemannian metrics, global analysis. Proc. Symp. Pure Math. 15, 11–40 (1968)

    Article  Google Scholar 

  13. Eguchi, T., Hanson, A.J.: Self-dual solutions to Euclidean gravity. Ann. Phys. 120, 82–106 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gursky, M.: Four-manifolds with \(\delta W^+ = 0\) and Einstein constants on the sphere. Math. Ann. 318, 417–43 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gursky, M., LeBrun, C.: On Einstein manifolds of positive sectional curvature. Ann. Glob. Anal. Geom. 17, 315–328 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hitchin, N.: Compact four-dimensional Einstein manifolds. J. Differ. Geom. 9, 435–441 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hitchin, N.: Einstein metrics and the eta-invariant. Bull. U. M. I. 11–B, 92–105 (1997)

    MathSciNet  MATH  Google Scholar 

  18. Jensen, G.R.: Einstein metrics on principal fibre bundles. J. Diff. Geom. 8, 599–614 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  19. Koiso, N.: Non-deformability of Einstein metrics. Osaka J. Math. 15, 419–433 (1978)

    MathSciNet  MATH  Google Scholar 

  20. Koiso, N.: Einstein metrics and complex structures. Invent. Math. 73, 71–106 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lawson, T.: Splitting \(S^4\) on \(\mathbb{RP}^2\) via the branched cover of \(\mathbb{CP}^2\) over \(S^4\). Proc. Am. Math. Soc. 86, 328–330 (1982)

    Google Scholar 

  22. LeBrun, C.: Einstein metrics and Mostow rigidity. Math. Res. Lett. 2, 1–8 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  23. LeBrun, C.: Four-manifolds without Einstein metrics. Math. Res. Lett. 3, 133–147 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lee, J., Parker, T.: The Yamabe problem. Bull. Am. Math. Soc. 17, 37–81 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  25. Nakajima, H.: Hausdorff convergence of Einstein \(4\)-manifolds. J. Fac. Sci. Univ. Tokyo 35, 411–424 (1988)

    MathSciNet  MATH  Google Scholar 

  26. Nakajima, H.: Self-duality of ALE Ricci-flat \(4\)-manifolds and positive mass theorem, in recent topics in differential and analytic geometry. Adv. Stud. Pure Math. 18–I, 313–349 (1990)

    Google Scholar 

  27. Nakajima, H.: A convergence theorem for Einstein metrics and the ALE spaces. Am. Math. Soc. Transl. 160, 79–94 (1994)

    MATH  Google Scholar 

  28. Obata, M.: The conjectures on conformal transformations of Riemannian manifolds. J. Diff. Geom. 6, 247–258 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  29. Schoen, R.: Variational theory for the total scalar curvature functional for Riemannian metrics and related topics. In: Giaquinta, M. (ed.) Topics in Calculus of Variations, Lect. Notes in Math, vol. 1365, pp. 121–154. Springer, Berlin (1989)

    Google Scholar 

  30. Thorpe, J.: Some remarks on the Gauss–Bonnet integral. J. Math. Mech. 18, 779–786 (1969)

    MathSciNet  MATH  Google Scholar 

  31. Yang, D.: Rigidity of Einstein \(4\)-manifolds with positive curvature. Invent. Math. 142, 435–450 (2000)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Anda Degeratu and Rafe Mazzeo for valuable discussions on the eta invariant, and Shouhei Honda for helpful discussions on convergence results of Riemannian manifolds with bounded Ricci curvature. They would also like to thank Matthew Gursky and Claude LeBrun for useful advice, and Gilles Carron and the anonymous referee for crucial comments.

Author information

Author notes

  1. Kazuo Akutagawa

    Present address: Department of Mathematics, Chuo University, Tokyo, 112-8551, Japan

Authors and Affiliations

  1. Department of Mathematics, Tokyo Institute of Technology, Tokyo, 152-8551, Japan

    Kazuo Akutagawa & Hisaaki Endo

  2. Mathematics Department, Indian Institute of Science, Bangalore, 560012, India

    Harish Seshadri

Authors
  1. Kazuo Akutagawa
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hisaaki Endo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Harish Seshadri
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kazuo Akutagawa.

Additional information

Communicated by F. C. Marques.

Kazuo Akutagawa: supported in part by the Grants-in-Aid for Scientific Research (B), Japan Society for the Promotion of Science, No. 24340008 and No. 18H01117.

Hisaaki Endo: supported in part by the Grants-in-Aid for Scientific Research (C), Japan Society for the Promotion of Science, No. 16K05142.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Akutagawa, K., Endo, H. & Seshadri, H. A gap theorem for positive Einstein metrics on the four-sphere. Math. Ann. 373, 1329–1339 (2019). https://doi.org/10.1007/s00208-018-1749-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-018-1749-x

Search

Navigation