Annals of Global Analysis and Geometry

, Volume 37, Issue 3, pp 263–274 | Cite as

On the asymptotic reduced volume of the Ricci flow

Article

Abstract

In this article, we consider two different monotone quantities defined for the Ricci flow and show that their asymptotic limits coincide for any ancient solutions. One of the quantities we consider here is Perelman’s reduced volume, while the other is the local quantity discovered by Ecker, Knopf, Ni, and Topping. This establishes a relation between these two monotone quantities.

Keywords

Ricci flow Reduced volume Ancient solution Monotonicity formula 

Mathematics Subject Classification (2000)

53C21 (primary) 35B40 58J35 

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References

  1. 1.
    Carrillo, J., Ni, L.: Sharp logarithmic Sobolev inequalities on gradient solitons and applications. arXiv:0806.2417Google Scholar
  2. 2.
    Chen B.-L.: Strong uniqueness of the Ricci flow. J. Differ. Geom. 82(2), 363–382 (2009)MATHGoogle Scholar
  3. 3.
    Chow, B., Lu, P., Ni, L.: Hamilton’s Ricci Flow. Graduate Studies in Mathematics, vol. 77. American Mathematical Society/Science Press, Providence/New York, (2006)Google Scholar
  4. 4.
    Chow, B., Chu, S.-C., Glickenstein, D., Guenther, C., Isenberg, J., Ivey, T., Knopf, D., Lu, P., Luo, F., Ni, L.: The Ricci Flow: Techniques and Applications. Part I. Geometric Aspects, Mathematical Surveys and Monographs, vol. 135. American Mathematical Society, Providence, RI, (2007)Google Scholar
  5. 5.
    Ecker, K.: Regularity Theory for Mean Curvature Flow. Progress in Nonlinear Differential Equations and their Applications, vol. 57. Birkhäuser Boston, Inc., Boston, MA (2004)Google Scholar
  6. 6.
    Ecker K., Knopf D., Ni L., Topping P.: Local monotonicity and mean value formulas for evolving Riemannian manifolds. J. Reine Angew. Math. 616, 89–130 (2008)MATHMathSciNetGoogle Scholar
  7. 7.
    Enders, J.: Reduced distance based at singular time in the Ricci flow. arXiv:0711.0558Google Scholar
  8. 8.
    Feldman M., Ilmanen T., Ni L.: Entropy and reduced distance for Ricci expanders. J. Geom. Anal. 15(1), 49–62 (2005)MATHMathSciNetGoogle Scholar
  9. 9.
    Lott J.: Optimal transport and Perelman’s reduced volume. Calc. Var. Partial Differ. Equ. 36(1), 49–84 (2009)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    McCann, R., Topping, P.: Ricci flow, entropy and optimal transportation, Amer. J. Math. (to appear)Google Scholar
  11. 11.
    Müller, R.: Differential Harnack Inequalities and the Ricci Flow. EMS Series of Lectures in Mathematics. European Mathematical Society (EMS), Zürich (2006)Google Scholar
  12. 12.
    Naber, A.: Noncompact shrinking 4-solitons with nonnegative curvature. arXiv:0710.5579Google Scholar
  13. 13.
    Ni, L.: The entropy formula for linear heat equation. J. Geom. Anal. 14(1), 87–100; Addenda: J. Geom. Anal. 14(2), 369–374 (2004)Google Scholar
  14. 14.
    Ni L.: A matrix Li–Yau–Hamilton estimate for Kähler-Ricci flow. J. Differ. Geom. 75(2), 303–358 (2007)MATHGoogle Scholar
  15. 15.
    Ni L.: Mean value theorems on manifolds. Asian J. Math. 11(2), 277–304 (2007)MATHMathSciNetGoogle Scholar
  16. 16.
    Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv:math/0211159Google Scholar
  17. 17.
    Topping, P.: \({\mathcal {L}}\) -optimal transportation for Ricci flow, J. Reine Angew. Math. (to appear)Google Scholar
  18. 18.
    Watson M.-A.: A theory of subtemperatures in several variables. Proc. Lond. Math. Soc. 26(3), 385–417 (1973)MATHCrossRefGoogle Scholar
  19. 19.
    Ye R.: On the l-function and the reduced volume of Perelman I. Trans. Am. Math. Soc. 360(1), 507–531 (2008)MATHCrossRefGoogle Scholar
  20. 20.
    Yokota T.: Perelman’s reduced volume and a gap theorem for the Ricci flow. Commun. Anal. Geom. 17(2), 227–263 (2009)MATHMathSciNetGoogle Scholar
  21. 21.
    Yokota, T.: A gap theorem for ancient solutions to the Ricci flow. Proceedings of the 1st MSJ-SI “Probabilistic Approach to Geometry” (to appear)Google Scholar
  22. 22.
    Zhang Z.-H.: On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc. 137(8), 2755–2759 (2009)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Graduate School of Pure and Applied SciencesUniversity of TsukubaTsukubaJapan

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