Advertisement

The Journal of Geometric Analysis

, Volume 28, Issue 4, pp 3718–3724 | Cite as

New Proofs of Perelman’s Theorem on Shrinking Breathers in Ricci Flow

  • Peng Lu
  • Yu ZhengEmail author
Article
  • 120 Downloads

Abstract

We give two new proofs of Perelman’s theorem that shrinking breathers of Ricci flow on closed manifolds are gradient Ricci solitons, using the fact that the singularity models of Type I solutions are shrinking gradient Ricci solitons and the fact that non-collapsed Type I ancient solutions have rescaled limits being shrinking gradient Ricci solitons.

Keywords

Ricci flow Shrinking breathers Type I solutions Gradient Ricci solitons 

Mathematics Subject Classification

53C44 

Notes

Acknowledgements

P. L. wants to thank Professor Li, Jiayu and School of Mathematical Sciences at the University of Science and Technology of China, where part of this work is carried out, for their warm hospitality during spring, 2017. The authors thank the referee for detailed suggestions which helped to improve the paper.

References

  1. 1.
    Cao, X.D., Zhang, Q.: The conjugate heat equation and ancient solutions of the Ricci flow. Adv. Math. 228, 2891–2919 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Enders, J., Müller, R., Topping, P.: On type-I singularities in Ricci flow. Comm. Anal. Geom. 19, 905–922 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Feldman, M., Ilmanen, T., Ni, L.: Entropy and reduced distance for Ricci expanders. J. Geom. Anal. 15, 49–62 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Hamilton, R.: Eternal solutions to the Ricci flow. J. Diff. Geom. 38, 1–11 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ivey, T.: Ricci solitons on compact three-manifolds. Diff. Geom. Appl. 3, 301–307 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Naber, A.: Noncompact shrinking four solitons with nonnegative curvature. J. Reine Angew. Math. 645, 125–153 (2010)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159
  8. 8.
    Rimondi, M., Veronelli, G.: Extremals of log Sobolev inequality on non-compact manifolds and Ricci soliton structures. ArXiv:1605.09240
  9. 9.
    Zhang, Q.: A no breathers theorem for some noncompact Ricci flows. Asian J. Math. 18, 727–756 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of OregonEugeneUSA
  2. 2.Department of MathematicsEast China Normal UniversityShanghaiPeople’s Republic of China

Personalised recommendations