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The Journal of Geometric Analysis

, Volume 28, Issue 4, pp 3424–3457 | Cite as

Ricci Flow on a Class of Noncompact Warped Product Manifolds

  • Tobias Marxen
Article
  • 150 Downloads

Abstract

We consider the Ricci flow on noncompact \(n+1\)-dimensional manifolds M with symmetries, corresponding to warped product manifolds \(\mathbb {R}\times T^n\) with flat fibres. We show longtime existence and that the Ricci flow solution is of type III, i.e. the curvature estimate \(|{{\mathrm{Rm}}}|(p,t) \le C/t\) for some \(C > 0\) and all \(p \in M, t \in (1,\infty )\) holds. We also show that if M has finite volume, the solution collapses, i.e. the injectivity radius converges uniformly to 0 (as \(t \rightarrow \infty \)) while the curvatures stay uniformly bounded, and furthermore, the solution converges to a lower dimensional manifold. Moreover, if the (n-dimensional) volumes of hypersurfaces coming from the symmetries of M are uniformly bounded, the solution converges locally uniformly to a flat cylinder after appropriate rescaling and pullback by a family of diffeomorphisms. Corresponding results are also shown for the normalized (i.e. volume preserving) Ricci flow.

Keywords

Ricci flow Warped product Noncompact 

Mathematics Subject Classification

53C44 (primary) 58D19 (secondary) 

Notes

Acknowledgements

This research was mostly carried out as part of my doctoral thesis, and at this point I would like to cordially thank my advisor Klaus Ecker! Also many thanks to Ahmad Afuni, Richard Bamler, Theodora Bourni, Bernhard Brehm, Bernold Fiedler, Lutz Habermann, Adrian Hammerschmidt, Gerhard Huisken, Tom Ilmanen, Felix Jachan, Dan Knopf, Ananda Lahiri, Markus Röser, Andreas Savas-Halilaj, Lars Schäfer, Oliver Schnürer, Felix Schulze, Brian Smith and Elmar Vogt! I also thank the SFB 647 “Space–Time–Matter. Analytic and Geometric Structures” of the DFG (German Research Foundation) for financial support.

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Copyright information

© Mathematica Josephina, Inc. 2017

Authors and Affiliations

  1. 1.Freie Universität BerlinBerlinGermany

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