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Convergence Stability for Ricci Flow

  • Eric Bahuaud
  • Christine GuentherEmail author
  • James Isenberg
Article
  • 9 Downloads

Abstract

The principle of convergence stability for geometric flows is the combination of the continuous dependence of the flow on initial conditions, with the stability of fixed points. It implies that if the flow from an initial state \(g_0\) exists for all time and converges to a stable fixed point, then the flows of solutions that start near \(g_0\) also converge to fixed points. We show this in the case of the Ricci flow, carefully proving the continuous dependence on initial conditions. Symmetry assumptions on initial geometries are often made to simplify geometric flow equations. As an application of our results, we extend known convergence results to open sets of these initial data, which contain geometries with no symmetries.

Keywords

Stability of the Ricci flow Continuous dependence of the Ricci flow Analytic semigroups 

Mathematics Subject Classification

53C44 58J35 35K 

Notes

Acknowledgements

The authors would like to thank Dan Knopf, Jack Lee, Rafe Mazzeo, and Haotian Wu for helpful conversations related to this work. We are grateful to the anonymous referee who made several suggestions that improved this paper. This work was supported by Grants from the Simons Foundation (#426628, E. Bahuaud and #283083, C. Guenther). J. Isenberg was partially supported by the NSF Grant DMS-1263431. This work was initiated at the 2015 BIRS workshop Geometric Flows: Recent Developments and Applications (15w5148).

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Copyright information

© Mathematica Josephina, Inc. 2019

Authors and Affiliations

  1. 1.Department of MathematicsSeattle UniversitySeattleUSA
  2. 2.Department of MathematicsPacific UniversityForest GroveUSA
  3. 3.Department of MathematicsUniversity of OregonEugeneUSA

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