Advertisement

Mathematische Zeitschrift

, Volume 274, Issue 1–2, pp 471–481 | Cite as

On Wilking’s criterion for the Ricci flow

  • H. A. Gururaja
  • Soma Maity
  • Harish SeshadriEmail author
Article

Abstract

Wilking has recently shown that one can associate a Ricci flow invariant cone of curvature operators \(C(S)\), which are nonnegative in a suitable sense, to every \(Ad_{SO(n,\mathbb{C })}\) invariant subset \(S \subset \mathbf{so}(n,\mathbb{C })\). In this article we show that if \(S\) is an \(Ad_{SO(n,\mathbb{C })}\) invariant subset of \(\mathbf{so}(n,\mathbb{C })\) such that \(S\cup \{0\}\) is closed and \(C_+(S)\subset C(S)\) denotes the cone of curvature operators which are positive in the appropriate sense then one of the two possibilities holds: (a) The connected sum of any two Riemannian manifolds with curvature operators in \(C_+(S)\) also admits a metric with curvature operator in \(C_+(S)\) (b) The normalized Ricci flow on any compact Riemannian manifold \(M\) with curvature operator in \(C_+(S)\) converges to a metric of constant positive sectional curvature. We also point out that if \(S\) is an arbitrary \(Ad_{SO(n,\mathbb{C })}\) subset, then \(C(S)\) is contained in the cone of curvature operators with nonnegative isotropic curvature.

Keywords

Sectional Curvature Curvature Operator Compact Riemannian Manifold Invariant Subset Ricci Flow 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Berger, M.: Sur quelques varietes riemanniens suffisament pincees. Bull. Soc. Math. Fr. 88, 55–71 (1960)Google Scholar
  2. 2.
    Böhm, C., Wilking, B.: Manifolds with positive curvature operators are space forms. Ann. Math. 167(3), 1079–1097 (2008)zbMATHCrossRefGoogle Scholar
  3. 3.
    Brendle, S., Schoen, R.: Manifolds with \(1/4\)-pinched curvature are space forms. J. Am. Math. Soc. 22(1), 287–307 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Brendle, S., Schoen, R.: Classification of manifolds with weakly \( 1/4\) -pinched curvatures. Acta Math. 200, 1–13 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Brendle, S.: A general convergence result for the Ricci flow in higher dimensions. Duke Math. J. 145(3), 585–601 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Brendle, S.: Einstein manifolds with nonnegative isotropic curvature are locally symmetric. Duke Math. J. 151(1), 1–21 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Chen, X.: On Kähler manifolds with positive orthogonal bisectional curvature. Adv. Math. 215, 427–225 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Chen, X., Sun, S., Tian, G.: A note on Kähler–Ricci soliton. Int. Math. Res. Not. 17, 3328–3336 (2009)MathSciNetGoogle Scholar
  9. 9.
    Collingwood, D., McGovern, W.: Nilpotent Orbits in Semisimple Lie Algebras. Van Nostrand Reinhold, New York (1993)zbMATHGoogle Scholar
  10. 10.
    Gu, H., Zhang, Z.: An extension of Mok’s theorem on the generalized Frankel conjecture. Sci. China Math. 53, 1253–1264 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Hamilton, R.S.: Four-manifolds with positive curvature operator. J. Differ. Geom. 24(2), 153–179 (1986)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Lee, N.: Determination of the partial positivity of the curvature in symmetric spaces. Annali di Matematica pura ed applicata (IV) CLXXI, 107–129 (1996)CrossRefGoogle Scholar
  13. 13.
    Liu, X.S.: The partial positivity of the curvature in Riemannian symmetric spaces. Chin. Ann. Math. 29B(3), 317–332 (2008)CrossRefGoogle Scholar
  14. 14.
    Micallef, M., Wang, M.: Metrics with nonnegative isotropic curvature. Duke Math. J. 72(3), 649–672 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Seshadri, H.: Manifolds with nonnegative isotropic curvature. Commun. Anal. Geom. 4, 621–635 (2009)MathSciNetGoogle Scholar
  16. 16.
    Wilking, B.: A Lie algebraic approach to Ricci flow invariant curvature conditions and Harnack inequalities. arXiv:1011.3561Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of ScienceBangaloreIndia

Personalised recommendations