The Rigidity of S 3 × R Under Ancient Ricci Flow

Yongjia Zhang

Research output: Contribution to journalArticlepeer-review


In this paper we generalize the neck-stability theorem of Kleiner–Lott (Acta Math 219: 65–134, 2014) to a special class of four-dimensional nonnegatively curved Type I κ-solutions, namely, those whose asymptotic shrinkers are the standard cylinder S 3 × R. We use this stability result to prove a rigidity theorem: if a four-dimensional Type I κ-solution with nonnegative curvature operator has the standard cylinder S 3 × R as its asymptotic shrinker, then it is exactly the cylinder with its standard shrinking metric.

Original languageEnglish (US)
Pages (from-to)1136-1152
Number of pages17
JournalJournal of Geometric Analysis
Issue number2
StatePublished - Apr 15 2019
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2018, Mathematica Josephina, Inc.


  • Ancient solution
  • Cylinder rigidity
  • Neck stability
  • Ricci flow


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