Advertisement

Einstein Four-Manifolds with Sectional Curvature Bounded from Above

  • Zhuhong ZhangEmail author
Article
  • 18 Downloads

Abstract

Given an Einstein structure \({\bar{g}}\) with positive scalar curvature on a four-dimensional Riemannian manifold, that is \({\bar{R}}ic=\lambda {\bar{g}}\) for some positive constant \(\lambda \), a basic problem is to classify such Einstein 4-manifolds with positive or nonnegative sectional curvature. For convenience, the Ricci curvature is always normalized to be \({\bar{R}}ic=1\). In this paper, we firstly show that if the sectional curvature of \({\bar{g}}\) satisfies \({\bar{K}}\le \frac{\sqrt{3}}{2}\approx 0.866025\), then \({\bar{g}}\) must have nonnegative sectional curvature. Next, we prove a rigidity theorem of Einstein four-manifolds with nonnegative sectional curvature satisfying the additional condition that \({\bar{K}}_{ik}+s{\bar{K}}_{ij}\ge K_s\) for every orthonormal basis \(\{e_i\}\) with \({\bar{K}}_{ik}\ge {\bar{K}}_{ij}\), where s is some nonnegative constant. More precisely, we show that such Einstein manifolds must be isometric to either \(S^4\), or \(RP^4\), or \(CP^2\) (with standard metrics respectively). As a corollary, we obtain a rigidity result of Einstein four-manifolds with \({\bar{R}}ic=1\) and the sectional curvature satisfying the upper bound \({\bar{K}} \le M_2 \approx 0.750912\).

Keywords

Einstein manifold Ricci flow Sectional curvature Curvature pinching estimate 

Mathematics Subject Classification

53C24 53C25 

Notes

Acknowledgements

The author was partially supported by NSFC 11301191. He is grateful to Professor Huai-Dong Cao for encouragement and very helpful discussions.

References

  1. 1.
    Berger, M.: Sur quelques varietes riemanniennes suffisamment pincees. Bull. Soc. Math. Fr. 88, 57–71 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Berger, M.: Sur quelques varietes d’Einstein compacts. Ann. Mat. Pur. Appl. 53, 89–96 (1961)CrossRefzbMATHGoogle Scholar
  3. 3.
    Besse, A.L.: Einstein manifolds, volume 10 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results inMathematics and Related Areas (3)]. Springer, Berlin (1987)Google Scholar
  4. 4.
    Böhm, C., Wilking, B.: Manifolds with positive curvature operators are space forms. Ann. Math. 167(3), 1079–1097 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brendle, S.: Einstein manifolds with nonnegative isotropy curvature are locally symmetric. Duke. Math. J. 151(1), 1–21 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Brendle, S., Schoen, R.: Manifolds with \(1/4\)-pinched curvature are space forms. J. Amer. Math. Soc. 22(1), 287–307 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cao, X.D., Tran, H.: Einstein four-manifolds of pinched sectional curvature, arXiv:1612.06023v1, (2016)
  8. 8.
    Chen, B.L., Zhu, X.P.: Ricci flow with surgery on four-manifolds with positive isotropic curvature. J. Diff. Geom. 74, 177–264 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chen, B.L., Tang, S.H., Zhu, X.P.: Complete classification of compact four-manifolds with positive isotropic curvature. J. Differ. Geom. 91, 1–169 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Costa, E.: On Einstein four-manifolds. J. Geom. Phys. 51(2), 244–255 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gursky, M.J., LeBrun, C.: On Einstein manifolds of positive sectional curvature. Ann. Glob. Anal. Geom. 17, 315–328 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hamilton, R.S.: Three manifolds with positive Ricci curvature. J. Diff. Geom. 17, 255–306 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hamilton, R.S.: Four-manifolds with positive curvature operator. J. Diff. Geom. 24, 153–179 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hamilton, R.S.: Four manifolds with positive isotropy curvature. Comm. Anal. Geom. 5, 1–92 (1997). (or see, Collected Papers on Ricci Flow, Edited by H. D. Cao, B. Chow, S. C. Chu and S. T. Yau, International Press 2002)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hitchin, N.J.: On compact four-dimensional Einstein manifolds. J. Diff. Geom. 9, 435–442 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ribeiro Jr., E.: Rigidity of four-dimensional compact manifolds with harmonic Weyl tensor. Annali di Matematica Pura ed Applicata 195, 2171–2181 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Tachibana, S.: A theorem of Riemannian manifolds of positive curvature operator. Proc. Jpn. Acad. 50, 301–302 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Tsagas, G.: A relation between Killing tensor fields and negative pinched Riemannian manifolds. Proc. Am. Math. Soc. 22, 476–478 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Yang, D.G.: Rigidity of Einstein 4-manifolds with positive curvature. Invent. Math 142, 435–450 (2000)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesSouth China Normal UniversityGuangzhouPeople’s Republic of China

Personalised recommendations