Skip to main content
Log in

New monotonicity formulas for Ricci curvature and applications. I

  • Published:
Acta Mathematica

Abstract

We prove three new monotonicity formulas for manifolds with a lower Ricci curvature bound and show that they are connected to rate of convergence to tangent cones. In fact, we show that the derivative of each of these three monotone quantities is bounded from below in terms of the Gromov–Hausdorff distance to the nearest cone. The monotonicity formulas are related to the classical Bishop–Gromov volume comparison theorem and Perelman’s celebrated monotonicity formula for the Ricci flow. We will explain the connection between all of these.

Moreover, we show that these new monotonicity formulas are linked to a new sharp gradient estimate for the Green function that we prove. This is parallel to the fact that Perelman’s monotonicity is closely related to the sharp gradient estimate for the heat kernel of Li–Yau.

In [CM4] one of the monotonicity formulas is used to show uniqueness of tangent cones with smooth cross-sections of Einstein manifolds.

Finally, there are obvious parallelisms between our monotonicity and the positive mass theorem of Schoen–Yau and Witten.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Anderson, M. T., Kronheimer, P. B. & LeBrun, C., Complete Ricci-flat Kähler manifolds of infinite topological type. Comm. Math. Phys., 125 (1989), 637–642.

    Article  MathSciNet  MATH  Google Scholar 

  2. Bakry, D. & Ledoux, M., A logarithmic Sobolev form of the Li–Yau parabolic inequality. Rev. Mat. Iberoam., 22 (2006), 683–702.

    Article  MathSciNet  MATH  Google Scholar 

  3. Cheeger, J. & Colding, T. H., Lower bounds on Ricci curvature and the almost rigidity of warped products. Ann. of Math., 144 (1996), 189–237.

    Article  MathSciNet  MATH  Google Scholar 

  4. — On the structure of spaces with Ricci curvature bounded below. I. J. Differential Geom., 46 (1997), 406–480.

  5. Cheeger, J., Colding, T. H. & Minicozzi, W. P., II, Linear growth harmonic functions on complete manifolds with nonnegative Ricci curvature. Geom. Funct. Anal., 5 (1995), 948–954.

    Article  MathSciNet  MATH  Google Scholar 

  6. Cheng, S. Y. & Yau, S.-T., Differential equations on Riemannian manifolds and their geometric applications. Comm. Pure Appl. Math., 28 (1975), 333–354.

    Article  MathSciNet  MATH  Google Scholar 

  7. Colding, T. H., Shape of manifolds with positive Ricci curvature. Invent. Math., 124 (1996), 175–191.

    Article  MathSciNet  MATH  Google Scholar 

  8. — Large manifolds with positive Ricci curvature. Invent. Math., 124 (1996), 193–214.

  9. — Ricci curvature and volume convergence. Ann. of Math., 145 (1997), 477–501.

  10. Colding, T. H. & Minicozzi, W. P., II, Harmonic functions with polynomial growth. J. Differential Geom., 46 (1997), 1–77.

    MathSciNet  MATH  Google Scholar 

  11. — Large scale behavior of kernels of Schrödinger operators. Amer. J. Math., 119 (1997), 1355–1398.

  12. — Monotonicity and its analytic and geometric implications. To appear in Proc. Natl. Acad. Sci. USA. http://www.pnas.org/content/early/2012/07/31/1203856109.full.pdf.

  13. — On uniqueness of tangent cones of Einstein manifolds. Preprint, 2012. arXiv:1206.4929 [math.DG].

  14. Colding, T. H. & Naber, A., Sharp Hölder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications. Ann. of Math., 176 (2012), 1173–1229.

    Article  MathSciNet  MATH  Google Scholar 

  15. — Characterization of tangent cones of noncollapsed limits with lower Ricci bounds and applications. To appear in Geom. Funct. Anal.

  16. Gilbarg, D. & Serrin, J., On isolated singularities of solutions of second order elliptic differential equations. J. Anal. Math., 4 (1955/56), 309–340.

  17. Gromov, M., Metric Structures for Riemannian and non-Riemannian Spaces. Modern Birkhäuser Classics. Birkhäuser, Boston, MA, 2007.

  18. Gromov, M., Lafontaine, J. & Pansu, P., Structures métriques pour les variétés riemanniennes. Nathan, Paris, 1981.

    MATH  Google Scholar 

  19. Huisken, G., Asymptotic behavior for singularities of the mean curvature flow. J. Differential Geom., 31 (1990), 285–299.

    MathSciNet  MATH  Google Scholar 

  20. Li, P., Tam, L.-F. & Wang, J., Sharp bounds for the Green’s function and the heat kernel. Math. Res. Lett., 4 (1997), 589–602.

    MathSciNet  MATH  Google Scholar 

  21. Li, P. & Yau, S.-T., On the parabolic kernel of the Schrödinger operator. Acta Math., 156 (1986), 153–201.

    Article  MathSciNet  Google Scholar 

  22. Menguy, X., Noncollapsing examples with positive Ricci curvature and infinite topological type. Geom. Funct. Anal., 10 (2000), 600–627.

    Article  MathSciNet  MATH  Google Scholar 

  23. — Examples of nonpolar limit spaces. Amer. J. Math., 122 (2000), 927–937

  24. Ni, L., The entropy formula for linear heat equation. J. Geom. Anal., 14 (2004), 87–100.

    Article  MathSciNet  MATH  Google Scholar 

  25. — Addenda to: “The entropy formula for linear heat equation” [J. Geom. Anal., 14 (2004), 87–100]. J. Geom. Anal., 14 (2004), 369–374.

  26. — Mean value theorems on manifolds. Asian J. Math., 11 (2007), 277–304.

  27. — The large time asymptotics of the entropy, in Complex Analysis, Trends Math., pp. 301–306. Birkhäuser, Basel, 2010.

  28. Perelman, G., Construction of manifolds of positive Ricci curvature with big volume and large Betti numbers, in Comparison Geometry (Berkeley, CA, 1993–94), Math. Sci. Res. Inst. Publ., 30, pp. 157–163. Cambridge Univ. Press, Cambridge, 1997.

  29. — A complete Riemannian manifold of positive Ricci curvature with Euclidean volume growth and nonunique asymptotic cone, in Comparison Geometry (Berkeley, CA, 1993–94), Math. Sci. Res. Inst. Publ., 30, pp. 165–166. Cambridge Univ. Press, Cambridge, 1997.

  30. — The entropy formula for the Ricci flow and its geometric applications. Preprint, 2002. arXiv:math/0211159 [math.DG].

  31. Reifenberg, E. R., Solution of the Plateau Problem for m-dimensional surfaces of varying topological type. Acta Math., 104 (1960), 1–92.

    Article  MathSciNet  MATH  Google Scholar 

  32. Toro, T., Geometric conditions and existence of bi-Lipschitz parameterizations. Duke Math. J., 77 (1995), 193–227.

    Article  MathSciNet  MATH  Google Scholar 

  33. Varopoulos, N. T., The Poisson kernel on positively curved manifolds. J. Funct. Anal., 44 (1981), 359–380.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Tobias Holck Colding.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Colding, T.H. New monotonicity formulas for Ricci curvature and applications. I. Acta Math 209, 229–263 (2012). https://doi.org/10.1007/s11511-012-0086-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11511-012-0086-2

Navigation