Annals of Global Analysis and Geometry

, Volume 42, Issue 2, pp 267–277 | Cite as

On the asymptotic behavior of expanding gradient Ricci solitons

  • Chih-Wei ChenEmail author


Let (M, g, f) be an n-dimensional expanding gradient Ricci soliton with faster-than-quadratic-decay curvature, i.e., \({\lim_{{\rm dist}(O,x)\rightarrow\infty} |{\rm Sect}(x)|\cdot {\rm dist}(O,x)^2=0}\) . When M is simply connected at infinity and n ≥ 3, we show that its tangent cone at infinity must be a manifold and is isometric to \({\mathbb{R}^n}\) . Here, we also assume that M has only one end for the simplicity of the statement. A crucial step to gain the regularity of the tangent cone at infinity is to prove that the injectivity radius grows linearly. This can be achieved by combining the curvature assumption and a lower bound estimate of volume ratio of all geodesic balls, which is attained as Theorem 3. On the other hand, we also study the asymptotic volume ratio of non-steady gradient Ricci solitons under other weaker conditions.


Ricci flow Expanding soliton Curvature decay 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    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)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Cao H.-D.: Limits of solutions to the Kähler-Ricci flow. J. Diff. Geom. 45, 257–272 (1997)zbMATHGoogle Scholar
  3. 3.
    Cao H.-D., Zhou D.T.: On complete gradient shrinking Ricci solitons. J. Diff. Geom. 2, 175–186 (2010)MathSciNetGoogle Scholar
  4. 4.
    Cao, H.-D., Chen, B.-L., Zhu, X.-P.: Recent developments on Hamilton’s Ricci flow. Surv. Diff. Geom. XII:47–112 (2008)Google Scholar
  5. 5.
    Carrillo J., Ni L.: Sharp logarithmic Sobolev inequalities on gradient solitons and applications. Comm. Anal. Geom. 17(4), 721–753 (2009)zbMATHGoogle Scholar
  6. 6.
    Catino G., Mantegazza C.: Evolution of the Weyl tensor under the Ricci flow. Ann. Inst. Fourier. 61(4), 1407–1435 (2011)zbMATHCrossRefGoogle Scholar
  7. 7.
    Cheeger J., Colding T.H.: Lower bounds on Ricci curvature and the almost rigidity of warped products. Ann. Math. 144, 189–237 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Cheeger J., Gromov M., Taylor M.: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds. J. Diff. Geom. 17, 15–53 (1982)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Chen B.-L.: Strong uniqueness of the Ricci flow. J. Diff. Geom. 82, 363–382 (2009)zbMATHGoogle Scholar
  10. 10.
    Chen B.-L., Zhu X.-P.: Complete Riemannian manifolds with pointwise pinched curvature. Invent. Math. 140, 423–452 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Chen, C.-W., Dereulle, A.: Expanding gradient Ricci soliton with finite asymptotic curvature ratio arXiv:1108.1468vl (2011)Google Scholar
  12. 12.
    Cheng S.Y., Li P., Yau S.-T.: On the upper estimate of the heat kernel of complete riemannian manifold. Am. J. Math. 103(5), 1021–1063 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Chow B., Lu P., Ni L.: Hamilton’s Ricci flow. Graduate Studies in Mathematics, vol. 77. American Mathematical Society, Providence, RI (2006)Google Scholar
  14. 14.
    Chow, B., Lu, P., Yang, B.: A lower bound for the scalar curvature of noncompact nonflat Ricci shrinkers. arXiv:1102.0548v1 (2011)Google Scholar
  15. 15.
    Chow, B., Lu, P., Yang, B.: A necessary and sufficient condition for Ricci shrinkers to have positive AVR. Proc. Am. Math. Soc. doi: 10.1090/S0002-9939-2011-11173-0
  16. 16.
    Feldman M., Ilmanen T., Knopf D.: Rotationally symmetric shrinking and expanding gradient Kähler-Ricci solitons. J. Diff. Geom. 65, 169–209 (2003)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Gutperle, M., Headrick, M., Minwalla, S., Schomerus, V.: Space-time energy decreases under world-sheet RG flow. J. High Energy Phys. 1 (2003)Google Scholar
  18. 18.
    Hamilton R.S.: The Harnack estimate for the Ricci flow. J. Diff. Geom. 37, 225–243 (1993)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Hamilton R.S.: The formation of singularities in the Ricci flow. Surv. Diff. Geom. 2, 7–136 (1995)MathSciNetGoogle Scholar
  20. 20.
    Munteanu, O.: The volume growth of complete gradient shrinking Ricci solitons. arXiv:0904.0798v2 (2009)Google Scholar
  21. 21.
    Naber A.: Noncompact shrinking four solitons with nonnegative curvature. J. Reine Angew. Math. 645, 125–153 (2010)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Ni L.: Ancient solutions to Kähler-Ricci flow. Math. Res. Lett. 12(5-6), 633–653 (2005)zbMATHMathSciNetGoogle Scholar
  23. 23.
    Ni L., Wallach N.: On a classification of the gradient Ricci solitons. Math. Res. Lett. 15(5), 941–955 (2008)zbMATHMathSciNetGoogle Scholar
  24. 24.
    Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv:math/ 0211159v1 (2002)Google Scholar
  25. 25.
    Petersen P., Wylie W.: On the classification of gradient Ricci solitons. Geom. Topol. 14(4), 2277–2300 (2010)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Wylie W.: Complete shrinking Ricci solitons have finite fundamental group. Proc. Am. Math. Soc. 136(5), 1803–1806 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Zhang Z.-H.: Gradient shrinking Ricci solitons with vanishing Weyl tensor. Pac. J. Math. 242(1), 189–200 (2009)zbMATHCrossRefGoogle Scholar
  28. 28.
    Zhang Z.-H.: On the completeness of gradient Ricci solitons. Proc. Am. Math. Soc. 137(8), 2755–2759 (2009)zbMATHCrossRefGoogle Scholar
  29. 29.
    Zhang S.J.: On a sharp volume estimate for gradient Ricci solitons with scalar curvature bounded below. Acta Math. Sin. 27(5), 871–882 (2011)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Department of MathematicsNational Taiwan UniversityTaipei CityTaiwan
  2. 2.Institut FourierUniversité Joseph FourierSaint Martin d’HèresFrance

Personalised recommendations