manuscripta mathematica

, 127:345

Ricci solitons: the equation point of view

  • Manolo Eminenti
  • Gabriele La Nave
  • Carlo Mantegazza
Article

Abstract

We discuss some classification results for Ricci solitons, that is, self similar solutions of the Ricci Flow. New simpler proofs of some known results will be presented. In detail, we will take the equation point of view, trying to avoid the tools provided by considering the dynamic properties of the Ricci flow.

Mathematics Subject Classification (2000)

53C 58J 

References

  1. 1.
    Baird P., Danielo L.: Three-dimensional Ricci solitons which project to surfaces. J. Reine Angew. Math. 608, 65–91 (2007)MATHMathSciNetGoogle Scholar
  2. 2.
    Böhm, C., Wilking, B.: Manifolds with positive curvature operators are space forms. Ann. Math. (2) (to appear)Google Scholar
  3. 3.
    Bryant, R.L.: Gradient Kähler Ricci solitons, ArXiv preprint server. http://arxiv.org (2004)
  4. 4.
    Cao, H.-D.: Existence of gradient Kähler-Ricci solitons, Elliptic and parabolic methods in geometry (Minneapolis, MN, 1994). A K Peters, Wellesley, pp. 1–16 (1996)Google Scholar
  5. 5.
    Cao H.-D.: Geometry of Ricci solitons. Chin. Ann. Math. Ser. B 27(2), 121–142 (2006)MATHCrossRefGoogle Scholar
  6. 6.
    Cao, H.-D., Chow, B., Chu, S.C., Yau, S.-T. (eds): Collected papers on Ricci flow. Ser Geom Topol vol. 37. International Press, Somerville (2003)Google Scholar
  7. 7.
    Cao X.: Compact gradient shrinking Ricci solitons with positive curvature operator. J. Geom. Anal. 17(3), 425–433 (2007)MATHMathSciNetGoogle Scholar
  8. 8.
    Cao, X., Wang, B.: On locally conformally flat gradient shrinking Ricci solitons, preprint (2008)Google Scholar
  9. 9.
    Chen X., Lu P., Tian G.: A note on uniformization of Riemann surfaces by Ricci flow. Proc. Am. Math. Soc. 134(11), 3391–3393 (2006) (electronic)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Chow, B., Knopf, D.: The Ricci flow: an introduction, mathematical surveys and monographs, vol. 110. American Mathematical Society, Providence (2004)Google Scholar
  11. 11.
    Derdzinski A.: A Myers-type theorem and compact Ricci solitons. Proc. AMS 134, 3645–3648 (2006)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Derdzinski, A.: Compact Ricci solitons (in preparation)Google Scholar
  13. 13.
    Fang, F., Man, J., Zhang, Z.: Complete gradient shrinking Ricci solitons have finite topological type, ArXiv Preprint Server. http://arxiv.org (2007)
  14. 14.
    Feldman M., Ilmanen T., Knopf D.: Rotationally symmetric shrinking and axpanding gradient K ähler–Ricci solitons. J. Diff. Geom. 65, 169–209 (2003)MATHMathSciNetGoogle Scholar
  15. 15.
    Fernández-López M., García-Río E.: A remark on compact Ricci solitons. Math. Ann. 340(4), 893–896 (2008)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Gallot S., Hulin D., Lafontaine J.: Riemannian geometry. Springer, Heidelberg (1990)MATHGoogle Scholar
  17. 17.
    Hamilton R.S.: Four-manifolds with positive curvature operator. J. Diff. Geom. 24(2), 153–179 (1986)MATHMathSciNetGoogle Scholar
  18. 18.
    Hamilton, R.S.: The Ricci flow on surfaces, Mathematics and general relativity (Santa Cruz, CA, 1986), Contemp. Math., vol. 71. American Mathematical Society, Providence, pp. 237–262 (1988)Google Scholar
  19. 19.
    Ivey T.: Ricci solitons on compact three-manifolds. Diff. Geom. Appl. 3(4), 301–307 (1993)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Koiso, N.: On rotationally symmetric Hamilton’s equation for Kähler–Einstein metrics. Recent topics in differential and analytic geometry, Adv. Stud. Pure Math., vol. 18. Academic Press, Boston, pp. 327–337 (1990)Google Scholar
  21. 21.
    Lott, J.: On the long-time behavior of type-iii Ricci flow solutions, ArXiv Preprint Server. http://arxiv.org (2005)
  22. 22.
    Naber, A.: Noncompact shrinking 4-solitons with nonnegative curvature, ArXiv Preprint Server. http://arxiv.org (2007)
  23. 23.
    Ni, L., Wallach, N.: On 4-dimensional gradient shrinking solitons, ArXiv Preprint Server. http://arxiv.org (2007)
  24. 24.
    Ni, L., Wallach, N.: On a classification of the gradient shrinking solitons, ArXiv Preprint Server. http://arxiv.org (2007)
  25. 25.
    Perelman, G. (2002) The entropy formula for the Ricci flow and its geometric applications, ArXiv Preprint Server. http://arxiv.org (2002)
  26. 26.
    Petersen, P., Wylie, W. (2007) On the classification of gradient Ricci solitons, ArXiv Preprint Server. http://arxiv.org (2007)
  27. 27.
    Rothaus O.S.: Logarithmic Sobolev inequalities and the spectrum of Schrödinger operators. J. Funct. Anal. 42(1), 110–120 (1981)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Tachibana S.: A theorem of Riemannian manifolds of positive curvature operator. Proc. Jpn. Acad. 50, 301–302 (1974)MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Manolo Eminenti
    • 1
  • Gabriele La Nave
    • 2
  • Carlo Mantegazza
    • 1
  1. 1.Scuola Normale Superiore PisaPisaItaly
  2. 2.Yeshiva UniversityNew YorkUSA

Personalised recommendations