Evolution equations in Riemannian geometry

Special Feature: The Takagi Lectures

Abstract

A fundamental question in Riemannian geometry is to find canonical metrics on a given smooth manifold. In the 1980s, R.S. Hamilton proposed an approach to this question based on parabolic partial differential equations. The goal is to start from a given initial metric and deform it to a canonical metric by means of an evolution equation. There are various natural evolution equations for Riemannian metrics, including the Ricci flow and the conformal Yamabe flow. In this survey, we discuss the global behavior of the solutions to these equations. In particular, we describe how these techniques can be used to prove the Differentiable Sphere Theorem.

Keywords and phrases

53C44 (primary) 35K55 53C21 58J35 (secondary) 

Mathematics Subject Classification (2010)

scalar curvature sectional curvature Yamabe problem Ricci flow Sphere Theorem 

References

  1. 1.
    Andrews B., Nguyen H.: Four-manifolds with 1/4-pinched flag curvatures. Asian J. Math. 13, 251–270 (2009)MathSciNetMATHGoogle Scholar
  2. 2.
    Aubin T.: Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire. J. Math. Pures Appl., (9) 55L, 269–296 (1976)MathSciNetGoogle Scholar
  3. 3.
    A. Bahri, Proof of the Yamabe conjecture, without the positive mass theorem, for locally conformally flat manifolds, In: Einstein Metrics and Yang–Mills Connections, (eds. T. Mabuchi and S. Mukai), Lecture Notes in Pure and Appl. Math., 145, Marcel Dekker, New York, 1993, pp. 1–26.Google Scholar
  4. 4.
    Berger M.: Les variétés Riemanniennes 1/4-pincées. Ann. Scuola Norm. Sup. Pisa (3) 14, 161–170 (1960)MathSciNetMATHGoogle Scholar
  5. 5.
    Besson G., (2006) Preuve de la conjecture de Poincaré en déformant la métrique par la courbure de Ricci (d’après G. Perelman), Séminaire Bourbaki, 2004/2005. Astérisque 307: 309–347Google Scholar
  6. 6.
    Böhm C., Wilking B.: Manifolds with positive curvature operator are space forms. Ann. of Math. (2) 167, 1079–1097 (2008)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Bourguignon J.-P., Ricci curvature and Einstein metrics, In: Global Differential Geometry and Global Analysis, Lecture Notes in Math., 838, Springer-Verlag, 1981, pp. 42–63Google Scholar
  8. 8.
    Brendle S.: Convergence of the Yamabe flow for arbitrary initial energy. J. Differential Geom. 69, 217–278 (2005)MathSciNetMATHGoogle Scholar
  9. 9.
    Brendle S.: A short proof for the convergence of the Yamabe flow on S n. Pure Appl. Math. Q. 3, 499–512 (2007)MathSciNetMATHGoogle Scholar
  10. 10.
    Brendle S.: Convergence of the Yamabe flow in dimension 6 and higher. Invent. Math. 170, 541–576 (2007)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Brendle S.: Blow-up phenomena for the Yamabe equation. J. Amer. Math. Soc. 21, 951–979 (2008)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Brendle S.: A general convergence result for the Ricci flow in higher dimensions. Duke Math. J. 145, 585–601 (2008)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Brendle S.: A generalization of Hamilton’s differential Harnack inequality for the Ricci flow. J. Differential Geom. 82, 207–227 (2009)MathSciNetMATHGoogle Scholar
  14. 14.
    S. Brendle, Ricci Flow and the Sphere Theorem, Grad. Stud. Math., 111, Amer. Math. Soc., Providence, RI, 2010.Google Scholar
  15. 15.
    Brendle S., Huisken G., Sinestrari C.: Ancient solutions to the Ricci flow with pinched curvature. Duke Math. J. 158, 537–551 (2011)MATHCrossRefGoogle Scholar
  16. 16.
    Brendle S., Marques F.C.: Blow-up phenomena for the Yamabe equation II. J. Differential Geom. 81, 225–250 (2009)MathSciNetMATHGoogle Scholar
  17. 17.
    Brendle S., Schoen R.: Classification of manifolds with weakly 1/4-pinched curvatures. Acta Math. 200, 1–13 (2008)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Brendle S., Schoen R.: Manifolds with 1/4-pinched curvature are space forms. J. Amer. Math. Soc. 22, 287–307 (2009)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Brendle S., Schoen R.: Curvature, sphere theorems, and the Ricci flow. Bull. Amer. Math. Soc. (N.S.) 48, 1–32 (2011)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Chen H.: Pointwise 1/4-pinched 4-manifolds. Ann. Global Anal. Geom. 9, 161–176 (1991)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Chen X.: Weak limits of Riemannian metrics in surfaces with integral curvature bounds. Calc. Var. Partial Differential Equations 6, 189–226 (1998)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Chen X.: Calabi flow in Riemann surface revisited: a new point of view. Internat. Math. Res. Notices 2001, 275–297 (2001)MATHCrossRefGoogle Scholar
  23. 23.
    Chen X., Lu P., Tian G.: A note on uniformization of Riemann surfaces by Ricci flow. Proc. Amer. Math. Soc. 134, 3391–3393 (2006)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Chow B.: The Ricci flow on the 2-sphere. J. Differential Geom. 33, 325–334 (1991)MathSciNetMATHGoogle Scholar
  25. 25.
    Chow B.: The Yamabe flow on locally conformally flat manifolds with positive Ricci curvature. Comm. Pure Appl. Math. 45, 1003–1014 (1992)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Daskalopoulos P., Hamilton R.S.: Geometric estimates for the logarithmic fast diffusion equation. Comm. Anal. Geom. 12, 143–164 (2004)MathSciNetMATHGoogle Scholar
  27. 27.
    DeTurck D.M.: Deforming metrics in the direction of their Ricci tensors. J. Differential Geom. 18, 157–162 (1983)MathSciNetMATHGoogle Scholar
  28. 28.
    Ecker K.: Heat equations in geometry and topology. Jahresber. Deutsch. Math.-Verein. 110, 117–141 (2008)MathSciNetMATHGoogle Scholar
  29. 29.
    Fraser A.M.: Fundamental groups of manifolds with positive isotropic curvature. Ann. of Math. (2) 158, 345–354 (2003)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Hamilton R.S.: Three-manifolds with positive Ricci curvature, J. Differential Geom. 17, 255–306 (1982)MathSciNetMATHGoogle Scholar
  31. 31.
    Hamilton R.S.: Four-manifolds with positive curvature operator. J. Differential Geom. 24, 153–179 (1986)MathSciNetMATHGoogle Scholar
  32. 32.
    Hamilton R.S.: The Ricci flow on surfaces. Contemp. Math. 71, 237–262 (1988)MathSciNetGoogle Scholar
  33. 33.
    R.S. Hamilton, Lectures on geometric flows, unpublished manuscript, 1989.Google Scholar
  34. 34.
    Hamilton R.S.: The Harnack estimate for the Ricci flow. J. Differential Geom. 37, 225–243 (1993)MathSciNetMATHGoogle Scholar
  35. 35.
    R.S. Hamilton, The formation of singularities in the Ricci flow, In: Surveys in Differential Geometry, vol. II, Int. Press, Cambridge, MA, 1995, pp. 7–136.Google Scholar
  36. 36.
    Hamilton R.S.: Four-manifolds with positive isotropic curvature. Comm. Anal. Geom. 5, 1–92 (1997)MathSciNetMATHGoogle Scholar
  37. 37.
    Huisken G.: Ricci deformation of the metric on a Riemannian manifold. J. Differential Geom. 21, 47–62 (1985)MathSciNetMATHGoogle Scholar
  38. 38.
    Klingenberg W.: Über Riemannsche Mannigfaltigkeiten mit positiver Krümmung. Comment. Math. Helv. 35, 47–54 (1961)MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Leeb B.: Geometrization of 3-dimensional manifolds and Ricci flow: on Perelman’s proof of the conjectures of Poincaré and Thurston. Boll. Unione Mat. Ital. (9) 1, 41–55 (2008)MathSciNetMATHGoogle Scholar
  40. 40.
    C. Margerin, Pointwise pinched manifolds are space forms, In: Geometric Measure Theory and the Calculus of Variations, Arcata, 1984, Proc. Sympos. Pure Math., 44, Amer. Math. Soc., Providence, RI, 1986, pp. 307–328.Google Scholar
  41. 41.
    Margerin C.: A sharp characterization of the smooth 4-sphere in curvature terms. Comm. Anal. Geom. 6, 21–65 (1998)MathSciNetMATHGoogle Scholar
  42. 42.
    Micallef M.J., Moore J.D.: Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes. Ann. of Math. (2) 127, 199–227 (1988)MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    S. Nishikawa, Deformation of Riemannian metrics and manifolds with bounded curvature ratios, In: Geometric Measure Theory and the Calculus of Variations, Arcata, 1984, Proc. Sympos. Pure Math., 44, Amer. Math. Soc., Providence, RI, 1986, pp. 343–352.Google Scholar
  44. 44.
    G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv:math/0211159.Google Scholar
  45. 45.
    G. Perelman, Ricci flow with surgery on three-manifolds, arXiv:math/0303109.Google Scholar
  46. 46.
    G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, arXiv:math/0307245.Google Scholar
  47. 47.
    Schoen R.: Conformal deformation of a Riemannian metric to constant scalar curvature. J. Differential Geom. 20, 479–495 (1984)MathSciNetMATHGoogle Scholar
  48. 48.
    R. Schoen, On the number of constant scalar curvature metrics in a conformal class, In: Differential Geometry, (eds. H.B. Lawson, Jr., and K. Tenenblat), Pitman Monogr. Surveys Pure Appl. Math., 52, Longman Sci. Tech., 1991, pp. 311–320.Google Scholar
  49. 49.
    Schoen R., Yau S.-T.: On the proof of the positive mass conjecture in general relativity. Comm. Math. Phys. 65, 45–76 (1979)MathSciNetMATHCrossRefGoogle Scholar
  50. 50.
    Schoen R., Yau S.-T.: Conformally flat manifolds, Kleinian groups. and scalar curvature. Invent. Math. 92, 47–71 (1988)MathSciNetMATHCrossRefGoogle Scholar
  51. 51.
    Schwetlick H., Struwe M.: Convergence of the Yamabe flow for large energies. J. Reine Angew. Math. 562, 59–100 (2003)MathSciNetMATHCrossRefGoogle Scholar
  52. 52.
    Simon L.: Asymptotics for a class of non-linear evolution equations, with applications to geometric problems. Ann. of. Math. (2) 118, 525–571 (1983)MathSciNetMATHCrossRefGoogle Scholar
  53. 53.
    Struwe M.: A global compactness result for elliptic boundary value problems involving limiting nonlinearities. Math. Z. 187, 511–517 (1984)MathSciNetMATHCrossRefGoogle Scholar
  54. 54.
    Struwe M.: Curvature flows on surfaces. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1, 247–274 (2002)MathSciNetMATHGoogle Scholar
  55. 55.
    Trudinger N.S.: Remarks concerning the conformal deformation of Riemannian structures on compact manifolds. Ann. Scuola Norm. Sup. Pisa (3) 22, 265–274 (1968)MathSciNetMATHGoogle Scholar
  56. 56.
    J. Wolf, Spaces of Constant Curvature. Fifth ed., Publish or Perish, Houston, TX, 1984.Google Scholar
  57. 57.
    Yamabe H.: On a deformation of Riemannian structures on compact manifolds. Osaka Math J. 12, 21–37 (1960)MathSciNetMATHGoogle Scholar
  58. 58.
    Ye R.: Global existence and convergence of Yamabe flow. J. Differential Geom. 39, 35–50 (1994)MathSciNetMATHGoogle Scholar

Copyright information

© The Mathematical Society of Japan and Springer Japan 2011

Authors and Affiliations

  1. 1.Department of MathematicsStanford UniversityStanfordUSA

Personalised recommendations