Annals of Global Analysis and Geometry

, Volume 42, Issue 2, pp 279–285 | Cite as

Integral Ricci curvature bounds along geodesics for nonexpanding gradient Ricci solitons

  • Bennett ChowEmail author
  • Peng Lu
  • Bo Yang


Following Li and Yau (Acta Math 156:153–201 1986) and similar to Perelman (The entropy formula for the Ricci flow and its geometric applications), we define an energy functional \({\mathcal{J}}\) associated to a smooth function \({\phi}\) on a complete Riemannian manifold. As an application, we deduce integral Ricci curvature upper bounds along modified geodesics for complete steady and shrinking gradient Ricci solitons.


Ricci soliton Ricci flow Ricci curvature 

Mathematics Subject Classification (2000)

53C44 58J35 53C22 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Angenent S., Knopf D.: Precise asymptotics of the Ricci flow neckpinch. Comm. Anal. Geom. 15(4), 773–844 (2007)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Angenent S., Isenberg J., Knopf D.: Formal matched asymptotics for degenerate Ricci flow neckpinches. Nonlinearity 24(8), 2265–2280 (2011)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Cao H.-D., Zhou D.: On complete gradient shrinking Ricci solitons. J. Differential Geom. 85, 175–186 (2010)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Carrillo J., Ni L.: Sharp logarithmic Sobolev inequalities on gradient solitons and applications. Comm. Anal. Geom. 17, 721–753 (2009)zbMATHGoogle Scholar
  5. 5.
    Chen B.-L.: Strong uniqueness of the Ricci flow. J. Differential Geom. 82, 363–382 (2009)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Fang F.-Q., Man J.-W., Zhang Z.-L.: Complete gradient shrinking Ricci solitons have finite topological type. C. R. Math. Acad. Sci. Paris 346, 653–656 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Feldman M., Ilmanen T., Ni L.: Entropy and reduced distance for Ricci expanders. J. Geom. Anal. 15, 49–62 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Fernández-Lopez M., García-Río E.: Maximum principles and gradient Ricci solitons. J. Differ Equa 251(1), 73–81 (2011)zbMATHCrossRefGoogle Scholar
  9. 9.
    Gu H.-L., Zhu X.-P.: The existence of type II singularities for the Ricci flow on S n+1. Comm. Anal. Geom. 16, 467–494 (2008)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Hamilton, R.S.: The Formation of Singularities in the Ricci Flow. Surveys in Differential Geometry, vol. II (Cambridge, MA, 1993), pp. 7–136. Internatonal Press, Cambridge, MA (1995)Google Scholar
  11. 11.
    Li P., Yau S.-T.: On the parabolic kernel of the Schrödinger operator. Acta Math. 156, 153–201 (1986)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Munteanu, O., Sesum, N.: On gradient Ricci solitons. arXiv:0910.1105v2 to appear in J. Geom. Anal.Google Scholar
  13. 13.
    Munteanu O., Wang J.: Smooth metric measure spaces with non-negative curvature. Comm. anal. Geom. 19(3), 451–486 (2011)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Munteanu, O., Wang, J.: Analysis of weighted Laplacian and applications to Ricci solitons. arXiv: 1112.3027vl. to appear in Comm. Anal. Geom.Google Scholar
  15. 15.
    Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv:math. DG/0211159Google Scholar
  16. 16.
    Pigola S., Rimoldi M., Setti A.G.: Remarks on non-compact gradient Ricci solitons. Math. Z. 268(3–4), 777–790 (2011)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Wu, P.: Remarks on gradient steady Ricci solitons. arXiv:1102.3018 to appear in J. Geom. Anal. doi: 10.1007/s12220-011-9243-7
  18. 18.
    Zhang S.: On a sharp volume estimate for gradient Ricci solitons with scalar curvature bounded below. Acta Math. Sin. (Engl. Ser.) 27(5), 871–882 (2011)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California, San DiegoLa JollaUSA
  2. 2.Department of MathematicsUniversity of OregonEugeneUSA

Personalised recommendations