Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

   
 
 

 

The canonical expanding soliton and Harnack inequalities for Ricci flow


Authors: Esther Cabezas-Rivas and Peter M. Topping
Journal: Trans. Amer. Math. Soc. 364 (2012), 3001-3021
MSC (2010): Primary 53C44
DOI: https://doi.org/10.1090/S0002-9947-2012-05391-8
Published electronically: February 13, 2012
MathSciNet review: 2888237
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We introduce the notion of Canonical Expanding Ricci Soliton, and use it to derive new Harnack inequalities for Ricci flow. This viewpoint also gives geometric insight into the existing Harnack inequalities of Hamilton and Brendle.


References [Enhancements On Off] (What's this?)

  • 1. S. Brendle and R. Schoen, Manifolds with $ 1/4$-pinched curvature are space forms. J. Amer. Math. Soc. 22 (2009) 287-307. MR 2449060 (2010a:53045)
  • 2. S. Brendle, A generalization of Hamilton's differential Harnack inequality for the Ricci flow. J. Differential Geom. 82 (2009), 207-227. MR 2504774 (2010d:53070)
  • 3. C. Böhm and B. Wilking, Manifolds with positive curvature operators are space forms. Ann. of Math. (2) 167 (2008), no. 3, 1079-1097. MR 2415394 (2009h:53146)
  • 4. E. Cabezas-Rivas and P.M. Topping, The Canonical Shrinking Soliton associated to a Ricci flow. Calc. Var. and PDE 43 (2012), 173-184.
  • 5. B. Chow and S.-C. Chu, A geometric interpretation of Hamilton's Harnack inequality for the Ricci flow. Math. Res. Lett. 2 (1995) 701-718. MR 1362964 (97f:53063)
  • 6. B. Chow and D. Knopf, New Li-Yau-Hamilton inequalities for the Ricci Flow via the space-time approach. J. Differential Geom. 60 (2002), 1-54. MR 1924591 (2003g:53116)
  • 7. B. Chow, P. Lu and L. Ni, Hamilton's Ricci flow. Graduate Studies in Math. 77, Amer. Math. Soc., Providence, RI (2006). MR 2274812 (2008a:53068)
  • 8. R. S. Hamilton, Three-manifolds with positive Ricci curvature. J. Differential Geom. 17 (1982) 255-306. MR 664497 (84a:53050)
  • 9. R. S. Hamilton, Four-manifolds with positive curvature operator. J. Differential Geom. 24 (1986), no. 2, 153-179. MR 862046 (87m:53055)
  • 10. R.S. Hamilton, The Harnack estimate for the Ricci flow. J. Differential Geom. 37 (1993) 225-243. MR 1198607 (93k:58052)
  • 11. R. S. Hamilton, The formation of singularities in the Ricci flow. Surveys in Differential Geometry, Vol. II (Cambridge, MA, 1993) 7-136, Internat. Press, Cambridge, MA, 1995. MR 1375255 (97e:53075)
  • 12. D. Knopf, Positivity of Ricci curvature under the Kähler-Ricci flow. Commun. Contemp. Math. 8 (2006), no. 1, 123-133. MR 2208813 (2006k:53114)
  • 13. S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol I. Interscience Publishers, New York-London (1963) xi+329 pp. MR 0152974 (27:2945)
  • 14. D. Máximo, Non-negative Ricci curvature on closed manifolds under Ricci flow. Proc. Amer. Math. Soc. 139 (2011), 675-685. MR 2736347
  • 15. R.J. McCann and P.M. Topping, Ricci flow, entropy and optimal transportation. Amer. J. Math. 132 (2010), 711-730. MR 2666905
  • 16. M.J. Micallef and J.D. Moore, Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes. Ann. of Math. (2) 127 (1988) 199-227. MR 924677 (89e:53088)
  • 17. G. Perelman, The entropy formula for the Ricci flow and its geometric applications. http://arXiv.org/abs/math/0211159v1 (2002).
  • 18. P.M. Topping, Lectures on the Ricci flow. London Math. Soc. Lecture Notes Series 325, Cambridge University Press, Cambridge, 2006. MR 2265040 (2007h:53105)
  • 19. P.M. Topping, $ \mathcal {L}$-optimal transportation for Ricci flow. J. Reine Angew. Math. 636 (2009) 93-122. MR 2572247
  • 20. P.M. Topping, Ricci Flow: The Foundations via Optimal Transportation. To appear in `Séminaires et Congrès.' S.M.F. (2009) http://www.warwick.ac.uk/~maseq
  • 21. B. Wilking, private communication (2009).
  • 22. B. Wilking, A Lie algebraic approach to Ricci Flow invariant curvature conditions and Harnack inequalities. To appear in J. Reine Angew. Math. arXiv:1011.3561

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 53C44

Retrieve articles in all journals with MSC (2010): 53C44


Additional Information

Esther Cabezas-Rivas
Affiliation: Mathematisches Institut, Fachbereich Mathematik und Informatik der Universität Münster, Einsteinstrasse 62, 48149 Münster, Germany

Peter M. Topping
Affiliation: Mathematics Institute, University of Warwick, Coventry, CV4 7AL, United Kingdom

DOI: https://doi.org/10.1090/S0002-9947-2012-05391-8
Received by editor(s): February 26, 2010
Received by editor(s) in revised form: June 4, 2010
Published electronically: February 13, 2012
Article copyright: © Copyright 2012 by the authors

American Mathematical Society