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A Compactness Theorem for Complete Ricci Shrinkers


We prove precompactness in an orbifold Cheeger–Gromov sense of complete gradient Ricci shrinkers with a lower bound on their entropy and a local integral Riemann bound. We do not need any pointwise curvature assumptions, volume or diameter bounds. In dimension four, under a technical assumption, we can replace the local integral Riemann bound by an upper bound for the Euler characteristic. The proof relies on a Gauss–Bonnet with cutoff argument.

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  1. AG

    Abresch U., Gromoll D.: On complete manifolds with nonnegative Ricci curvature. J. Amer. Math. Soc. 3(2), 355–374 (1990)

    MathSciNet  MATH  Article  Google Scholar 

  2. An

    Anderson M.: Ricci curvature bounds and Einstein metrics on compact manifolds. J. Amer. Math. Soc. 23, 455–490 (1989)

    Article  Google Scholar 

  3. AnC

    Anderson M., Cheeger J.: Diffeomorphism finiteness for manifolds with Ricci curvature and L n/2-norm of curvature bounded, Geom. Funct. Anal. 1(3), 231–252 (1991)

    MathSciNet  MATH  Article  Google Scholar 

  4. BKN

    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(2), 313–349 (1989)

    MathSciNet  MATH  Article  Google Scholar 

  5. BeBBMP

    L. Bessières, G. Besson, M. Boileau, S. Maillot, J. Porti, Geometrisation of 3-Manifolds, EMS Tracts in Mathematics 13, Zürich (2010).

  6. C

    H.-D. Cao, Recent progress on Ricci solitons, in “Recent Advances in Geometric Analysis, Advanced Lectures in Mathematics (ALM) 11, International Press, (2009).

  7. CS

    Cao H.-D., Sesum N.: A compactness result for Kähler Ricci solitons. Adv. Math. 211(2), 794–818 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  8. CZ

    Cao H.-D., Zhou D.: On complete gradient shrinking Ricci solitons. J. Diff. Geom. 85(2), 175–186 (2010)

    MathSciNet  MATH  Google Scholar 

  9. CZh

    H.-D. Cao, X.-P. Zhu, Hamilton-Perelman’s proof of the Poincaré conjecture and the geometrization conjecture, (2006); arXiv:math/0612069v1

  10. CaN

    Carrillo J., Ni L.: Sharp logarithmic Sobolev inequalities on gradient solitons and applications. Comm. Anal. Geom. 17(4), 721–753 (2009)

    MATH  Google Scholar 

  11. Ch1

    Cheeger J.: Finiteness theorems for Riemannian manifolds. Amer. J. Math. 92, 61–74 (1970)

    MathSciNet  MATH  Article  Google Scholar 

  12. Ch2

    Cheeger J.: Degeneration of Einstein metrics and metrics with special holonomy. Surveys Diff. Geom. VIII: 29(−73), 29–73 (2003)

    MathSciNet  Google Scholar 

  13. ChGT

    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(1), 15–53 (1982)

    MathSciNet  MATH  Google Scholar 

  14. CheW

    X. Chen, B. Wang, Space of Ricci flows (I), preprint (2009); arXiv:0902.1545v1

  15. ChoK

    B. Chow, D. Knopf, The Ricci Flow: An Introduction, Mathematical Surveys and Monographs 110, AMS, Providence (2004).

    Google Scholar 

  16. Cr

    Croke C.: Some isoperimetric inequalities and eigenvalue estimates. Ann. Sci. École Norm. Sup. (4) 13(4), 419–435 (1980)

    MathSciNet  MATH  Google Scholar 

  17. EH

    Eguchi T., Hanson A.: Gravitational instantons. Gen. Relativity Gravitation 11(5), 315–320 (1979)

    MathSciNet  Article  Google Scholar 

  18. FIK

    Feldman M., Ilmanen T., Knopf D.: Rotationally symmetric shrinking and expanding gradient Kähler-Ricci solitons. J. Diff. Geom. 65(2), 169–20 (2003)

    MathSciNet  MATH  Google Scholar 

  19. G

    P. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah–Singer Index Theorem, Math. Lect. Series 11, Publish or Perish (1984).

  20. GrW

    Greene R., Wu H.: Lipschitz convergence of Riemannian manifolds. Pacific J. Math. 131(1), 119–141 (1988)

    MathSciNet  MATH  Google Scholar 

  21. Gro

    M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, Progress in Mathematics 152, Birkhäuser Boston (1999).

  22. H1

    Hamilton R.: Three-manifolds with positive Ricci curvature. J. Diff. Geom. 17(2), 255–306 (1982)

    MathSciNet  MATH  Google Scholar 

  23. H2

    Hamilton R.: A compactness property for solutions of the Ricci flow. Amer. J. Math. 117, 545–572 (1995)

    MathSciNet  MATH  Article  Google Scholar 

  24. H3

    Hamilton R.: The formation of singularities in the Ricci flow. Surveys Diff. Geom. II: 7, 7–136 (1995)

    MathSciNet  Google Scholar 

  25. KL

    Kleiner B., Lott J.: Notes on Perelman’s papers. Geom. Topol. 12(5), 2587–2855 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  26. MT

    J. Morgan, G. Tian, Ricci Flow and the Poincaré Conjecture, Clay Mathematics Monographs 3, AMS, Cambridge, MA (2007).

  27. Mu

    O. Munteanu, The volume growth of complete gradient shrinking Ricci solitons, preprint (2009); arXiv:0904.0798v2

  28. MuS

    O. Munteanu, N. Sesum, On gradient Ricci solitons, preprint (2009); arXiv:0910.1105v1

  29. N

    Nakajima H.: Hausdorff convergence of Einstein 4-manifolds. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 35(2), 411–424 (1988)

    MathSciNet  MATH  Google Scholar 

  30. P

    G. Perelman, The entropy formula for the Ricci flow and its geometric applications, (2002); arXiv:math/0211159v1

  31. S

    Sibner L.: The isolated point singularity problem for the coupled Yang–Mills equations in higher dimensions. Math. Ann. 271(1), 125–131 (1985)

    MathSciNet  MATH  Article  Google Scholar 

  32. SoW

    J. Song, B. Weinkove, Contracting exceptional divisors by the Kähler-Ricci flow, preprint (2010); arXiv:1003.0718v1

  33. T

    Tian G.: On Calabi’s conjecture for complex surfaces with positive first Chern class. Invent. Math. 101(1), 101–172 (1990)

    MathSciNet  MATH  Article  Google Scholar 

  34. TV1

    Tian G., Viaclovsky J.: Bach-flat asymptotically locally Euclidean metrics. Invent. Math. 160(2), 357–415 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  35. TV2

    Tian G., Viaclovsky J.: Moduli spaces of critical Riemannian metrics in dimension four. Adv. Math. 196(2), 346–372 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  36. TV3

    Tian G., Viaclovsky J.: Volume growth, curvature decay, and critical metrics. Comment. Math. Helv. 83(4), 889–911 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  37. U

    Uhlenbeck K.: Removable singularities in Yang–Mills fields. Comm. Math. Phys. 83(1), 11–29 (1982)

    MathSciNet  MATH  Article  Google Scholar 

  38. V

    C. Villani, Optimal transport. Old and new, Grundlehren der Mathematischen Wissenschaften 338, Springer-Verlag (2009).

  39. W

    Weber B.: Convergence of compact Ricci Solitons. Int. Math. Res. Not. 1, 96–118 (2011)

    Google Scholar 

  40. WeW

    Wei G., Wylie W.: Comparison geometry for the Bakry–Emery Ricci tensor. J. Diff. Geom. 83(2), 337–405 (2009)

    MathSciNet  Google Scholar 

  41. Z

    Zhang X.: Compactness theorems for gradient Ricci solitons. J. Geom. Phys. 56(12), 2481–2499 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  42. Zh

    Zhang Z.H.: On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc. 137, 8–27552759 (2009)

    Google Scholar 

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Correspondence to Robert Haslhofer.

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Haslhofer, R., Müller, R. A Compactness Theorem for Complete Ricci Shrinkers. Geom. Funct. Anal. 21, 1091 (2011).

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Keywords and phrases

  • Ricci solitons
  • Ricci flow
  • Gauss–Bonnet with boundary

2010 Mathematics Subject Classification

  • 53C21
  • 53C23
  • 53C25
  • 53C44