Advertisement

The Journal of Geometric Analysis

, Volume 15, Issue 1, pp 49–62 | Cite as

Entropy and reduced distance for Ricci expanders

  • Michael Feldman
  • Tom Ilmanen
  • Lei Ni
Article

Abstract

Perelman has discovered two integral quantities, the shrinker entropy W and the (backward) reduced volume, that are monotone under the Ricci flow ∂gij/∂t = − 2Rij and constant on shrinking solitons. Tweaking some signs, we find similar formulae corresponding to the expanding case. The expanding entropy W+ is monotone on any compact Ricci flow and constant precisely on expanders; as in Perelman, it follows from a differential inequality for a Harnack-like quantity for the conjugate heat equation, and leads to functionals μ+ and v+. The forward reduced volume θ+ is monotone in general and constant exactly on expanders.

A natural conjecture asserts that g(t)/t converges as t → ∞ to a negative Einstein manifold in some weak sense (in particular ignoring collapsing parts). If the limit is known a-priori to be smooth and compact, this statement follows easily from any monotone quantity that is constant on expanders; these include vol(g)/tn/2 (Hamilton) and -λ (Perelman), as well as our new quantities. In general, we show that, if vol(g) grows like tn/2(maximal volume growth) then W+, θ+ and -λ remain bounded (in their appropriate ways) for all time. We attempt a sharp formulation of the conjecture.

Math Subject Classifications

58G11 

Key Words and Phrases

Entropy Ricci flow Ricci expanders asymptotic behavior reduced volume monotonicity Li-Yau-Hamilton inequality 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Anderson, M. T. The L2 structure of moduli spaces of Einstein metrics on 4-manifolds,Geom. Funct. Anal. 2, 29–89, (1992).CrossRefMathSciNetMATHGoogle Scholar
  2. [2]
    Angenent, S. B., Chopp, D., and Ilmanen, T. A computed example of nonuniqueness of mean curvature flow in ℝ3,Comm. Part. Diff. Eq. 20, 1937–1958, (1995).CrossRefMathSciNetGoogle Scholar
  3. [3]
    Barnes, I. and Ilmanen, T. unpublished computer study, (1994).Google Scholar
  4. [4]
    Cao, H.-D., Hamilton, R. S., and Ilmanen, T. Gaussian densities and stability for some Ricci solitons, math.DG/ 0404165, April (2004).Google Scholar
  5. [5]
    Cheeger, J. and Colding, T. H. On the structure of spaces with Ricci curvature bounded below. I,J. Differential Geom. 46, 406–480, (1997).MathSciNetMATHGoogle Scholar
  6. [6]
    Cheeger, J. and Tian, G. Collapsing and noncollapsing of Einstein 4-manifolds, in preparation, (2004).Google Scholar
  7. [7]
    Chow, B., Chu, S.-C., Lu, P., and Ni, L. Notes on Perelman’s articles on Ricci flow.Google Scholar
  8. [8]
    Chow, B. and Knopf, D. The Ricci flow, vol. I: An introduction, Math. Surveys and Monographs,AMS 110, (2004).Google Scholar
  9. [9]
    Daskalopoulos, P. Eternal solutions to the Ricci flow on ℝ2, preprint, (2004).Google Scholar
  10. [10]
    Feldman, M., Ilmanen, T., and Knopf, D. Rotationally symmetric shrinking and expanding gradient Kähler-Ricci solitons,J. Differential Geom. 65, 169–209,(2003).MathSciNetMATHGoogle Scholar
  11. [11]
    Hamilton, R. S. The Harnack estimate for the Ricci flow,J. Differential Geom. 37, 225–243, (1993).MathSciNetMATHGoogle Scholar
  12. [12]
    Hamilton, R. S. A compactness property for solutions of the Ricci flow,Amer. J. Math. 117, 545–572, (1995).CrossRefMathSciNetMATHGoogle Scholar
  13. [13]
    Hamilton, R. S. Non-singular solutions of Ricci flow on three-manifolds,Comm. Anal. Geom. 7, 695–729, (1999).MathSciNetMATHGoogle Scholar
  14. [14]
    Hong, M.-C. and Tian, G. Asymptotical behavior of the Yang-Mills flow and singular Yang-Mills connections,Math. Ann., to appear.Google Scholar
  15. [15]
    Huisken, G. Asymptotic behavior for singularities of the mean curvature flow,J. Differential Geom. 31, 285–299, (1990).MathSciNetMATHGoogle Scholar
  16. [16]
    Ilmanen, T. Elliptic regularization and partial regularity for motion by mean curvature,Mem. Amer. Math. Soc. #520, (1994).Google Scholar
  17. [17]
    Ilmanen, T. Singularities of mean curvature flow of surfaces, preprint, http://www.math.ethz.ch/~ ilmanen/papers/pub.htm, (1995).Google Scholar
  18. [18]
    Ilmanen, T. Lectures on mean curvature flow and related equations, Lecture Notes, ICTP, Trieste, http:// www.math.ethz.ch/~ilmanen/papers/pub.html,(1995).Google Scholar
  19. [19]
    Ilmanen, T. Notes on mean curvature flow, in preparation,(2004).Google Scholar
  20. [20]
    Li, P. and Yau, S.-T. On the parabolic kernel of the Schrödinger operator,Acta Math. 156, 153–201, (1986).CrossRefMathSciNetGoogle Scholar
  21. [21]
    Nash, J. Continuity of solutions of parabolic and elliptic equations,Amer. J. Math. 80, 935–954, (1958).CrossRefMathSciNetGoogle Scholar
  22. [22]
    Ni, L. The entropy formula for linear heat equation,J. Geom. Anal. 14(1), 87–100, (2004).MathSciNetMATHGoogle Scholar
  23. [23]
    Ni, L. Addenda to The entropy formula for linear heat equation,J. Geom. Anal. 14(2), 369–374, (2004).MathSciNetMATHGoogle Scholar
  24. [24]
    Ni, L. A new matrix Li-Yau-Hamilton inequality for Kahler-Ricci flow, submitted.Google Scholar
  25. [25]
    Perelman, G. The entropy formula for the Ricci flow and its geometric applications, math.DG/ 0211159, November (2002).Google Scholar
  26. [26]
    Perelman, G. Ricci flow with surgery on three-manifolds, math.DG/0303109, March (2003).Google Scholar
  27. [27]
    Sesum, N. Limiting behavior of the Ricci flow, preprint, (2004).Google Scholar
  28. [28]
    Uhlenbeck, K. Removable singularities in Yang-Mills fields,Comm. Math. Phys. 83, 11–29, (1982).CrossRefMathSciNetMATHGoogle Scholar
  29. [29]
    Weinkove, B. A complex Frobenuis theorem, multiplier ideal sheaves and Hermitian-Einstein metrics on stable bundles, preprint, (2003).Google Scholar
  30. [30]
    White, B. personal communication, (2000).Google Scholar

Copyright information

© Mathematica Josephina, Inc. 2005

Authors and Affiliations

  • Michael Feldman
    • 1
    • 2
  • Tom Ilmanen
    • 1
    • 2
  • Lei Ni
    • 3
  1. 1.Department of MathematicsUniversity of WisconsinMadison
  2. 2.Departement MathematikETH ZentrumZürichSwitzerland
  3. 3.Department of MathematicsUniversity of CaliforniaLa Jolla

Personalised recommendations