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


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.


Ricci flow Shrinking breathers Type I solutions Gradient Ricci solitons 

Mathematics Subject Classification




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.


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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

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