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The Journal of Geometric Analysis

, Volume 29, Issue 2, pp 1136–1152 | Cite as

The Rigidity of \(\mathbb {S}^3 \times \mathbb {R}\) Under Ancient Ricci Flow

  • Yongjia ZhangEmail author
Article
  • 132 Downloads

Abstract

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 \(\kappa \)-solutions, namely, those whose asymptotic shrinkers are the standard cylinder \(\mathbb {S}^3\times \mathbb {R}\). We use this stability result to prove a rigidity theorem: if a four-dimensional Type I \(\kappa \)-solution with nonnegative curvature operator has the standard cylinder \(\mathbb {S}^3\times \mathbb {R}\) as its asymptotic shrinker, then it is exactly the cylinder with its standard shrinking metric.

Keywords

Ricci flow Ancient solution Cylinder rigidity Neck stability 

Mathematics Subject Classification

53C44 

Notes

Acknowledgements

The author would like to thank his doctoral advisors, Professor Bennett Chow and Professor Lei Ni, for their constant support and priceless advices. The author also bids one of the referees all his gratitude, without whose patient refereeing this article could never have been improved thus far.

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Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaSan DiegoUSA

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