Advertisement

Inventiones mathematicae

, Volume 194, Issue 3, pp 731–764 | Cite as

Rotational symmetry of self-similar solutions to the Ricci flow

  • Simon Brendle
Article

Abstract

Let (M,g) be a three-dimensional steady gradient Ricci soliton which is non-flat and κ-noncollapsed. We prove that (M,g) is isometric to the Bryant soliton up to scaling. This solves a problem mentioned in Perelman’s first paper.

Keywords

Soliton Vector Field Scalar Curvature Ricci Soliton Ricci Flow 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

It is a pleasure to thank Professors Huai-Dong Cao, Gerhard Huisken, Sergiu Klainerman, Leon Simon, Brian White, for discussions. The author is grateful to Meng Zhu for comments on an earlier version of this paper.

References

  1. 1.
    Anderson, G., Chow, B.: A pinching estimate for solutions of the linearized Ricci flow system on 3-manifolds. Calc. Var. Partial Differ. Equ. 23, 1–12 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Brendle, S.: Ricci Flow and the Sphere Theorem. Graduate Studies in Mathematics, vol. 111. Am. Math. Soc., Providence (2010) zbMATHGoogle Scholar
  3. 3.
    Bryant, R.L.: Ricci flow solitons in dimension three with SO(3)-symmetries. Available at www.math.duke.edu/~bryant/3DRotSymRicciSolitons.pdf
  4. 4.
    Cao, H.D.: Recent progress on Ricci solitons. arXiv:0908.2006
  5. 5.
    Cao, H.D., Chen, Q.: On locally conformally flat gradient steady Ricci solitons. Trans. Am. Math. Soc. 364, 2377–2391 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cao, H.D., Catino, G., Chen, Q., Mantegazza, C., Mazzieri, L.: Bach flat gradient steady Ricci solitons. arXiv:1107.4591
  7. 7.
    Chen, B.L.: Strong uniqueness of the Ricci flow. J. Differ. Geom. 82, 363–382 (2009) zbMATHGoogle Scholar
  8. 8.
    Chen, X.X., Wang, Y.: On four-dimensional anti-self-dual gradient Ricci solitons. arXiv:1102.0358
  9. 9.
    Guo, H.: Area growth rate of the level surface of the potential function on the 3-dimensional steady Ricci soliton. Proc. Am. Math. Soc. 137, 2093–2097 (2009) CrossRefzbMATHGoogle Scholar
  10. 10.
    Gursky, M.: The Weyl functional, de Rham cohomology, and Kähler–Einstein metrics. Ann. Math. 148, 315–337 (1998) MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hamilton, R.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17, 255–306 (1982) MathSciNetzbMATHGoogle Scholar
  12. 12.
    Hamilton, R.: The formation of singularities in the Ricci flow. In: Surveys in Differential Geometry, vol. II, pp. 7–136. International Press, Somerville (1995) Google Scholar
  13. 13.
    Hawking, S.W., Ellis, G.F.R.: The Large Scale Structure of Space-Time. Cambridge University Press, Cambridge (1973) CrossRefzbMATHGoogle Scholar
  14. 14.
    Hopf, E.: On S. Bernstein’s theorem on surfaces z(x,y) of nonpositive curvature. Proc. Am. Math. Soc. 1, 80–85 (1950) MathSciNetzbMATHGoogle Scholar
  15. 15.
    Ionescu, A., Klainerman, S.: On the uniqueness of smooth stationary black holes in vacuum. Invent. Math. 175, 35–102 (2009) MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ivey, T.: New examples of complete Ricci solitons. Proc. Am. Math. Soc. 122, 241–245 (1994) MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kazdan, J., Warner, F.: Curvature functions for compact 2-manifolds. Ann. Math. 99, 14–47 (1974) MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lieberman, G.: Second Order Parabolic Differential Equations. World Scientific, River Edge (1996) CrossRefzbMATHGoogle Scholar
  19. 19.
    Morgan, J., Tian, G.: Ricci Flow and the Poincaré Conjecture. Am. Math. Soc., Providence (2007) zbMATHGoogle Scholar
  20. 20.
    Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv:math/0211159
  21. 21.
    Perelman, G.: Ricci flow with surgery on three-manifolds. arXiv:math/0303109
  22. 22.
    Perelman, G.: Finite extinction time for solutions to the Ricci flow on certain three-manifolds. arXiv:math/0307245
  23. 23.
    Simon, L.: Isolated singularities of extrema of geometric variational problems. In: Harmonic Mappsing and Minimal Immersions, Montecatini, 1984. Lectures Notes in Mathematics, vol. 1161, pp. 206–277 (1985) CrossRefGoogle Scholar
  24. 24.
    Simon, L., Solomon, B.: Minimal hypersurfaces asymptotic to quadratic cones in ℝn+1. Invent. Math. 86, 535–551 (1986) MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Struwe, M.: Curvature flows on surfaces. Ann. Sc. Norm. Super. Pisa, Ser. V 1, 247–274 (2002) MathSciNetzbMATHGoogle Scholar
  26. 26.
    Topping, P.: Lectures on the Ricci Flow. London Mathematical Society Lecture Notes Series, vol. 325. Cambridge University Press, Cambridge (2006) CrossRefzbMATHGoogle Scholar
  27. 27.
    Wang, X.J.: Convex solutions to the mean curvature flow. Ann. Math. 173, 1185–1239 (2011) CrossRefzbMATHGoogle Scholar
  28. 28.
    Zhang, Z.H.: On the completeness of gradient Ricci solitons. Proc. Am. Math. Soc. 137, 2755–2759 (2009) CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of MathematicsStanford UniversityStanfordUSA

Personalised recommendations