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Evolution equations in Riemannian geometry

  • Special Feature: The Takagi Lectures
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Japanese Journal of Mathematics Aims and scope

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.

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References

  1. Andrews B., Nguyen H.: Four-manifolds with 1/4-pinched flag curvatures. Asian J. Math. 13, 251–270 (2009)

    MathSciNet  MATH  Google Scholar 

  2. Aubin T.: Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire. J. Math. Pures Appl., (9) 55L, 269–296 (1976)

    MathSciNet  Google Scholar 

  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.

  4. Berger M.: Les variétés Riemanniennes 1/4-pincées. Ann. Scuola Norm. Sup. Pisa (3) 14, 161–170 (1960)

    MathSciNet  MATH  Google Scholar 

  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: 309–347

  6. Böhm C., Wilking B.: Manifolds with positive curvature operator are space forms. Ann. of Math. (2) 167, 1079–1097 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  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. 42–63

  8. Brendle S.: Convergence of the Yamabe flow for arbitrary initial energy. J. Differential Geom. 69, 217–278 (2005)

    MathSciNet  MATH  Google Scholar 

  9. Brendle S.: A short proof for the convergence of the Yamabe flow on S n. Pure Appl. Math. Q. 3, 499–512 (2007)

    MathSciNet  MATH  Google Scholar 

  10. Brendle S.: Convergence of the Yamabe flow in dimension 6 and higher. Invent. Math. 170, 541–576 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Brendle S.: Blow-up phenomena for the Yamabe equation. J. Amer. Math. Soc. 21, 951–979 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Brendle S.: A general convergence result for the Ricci flow in higher dimensions. Duke Math. J. 145, 585–601 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Brendle S.: A generalization of Hamilton’s differential Harnack inequality for the Ricci flow. J. Differential Geom. 82, 207–227 (2009)

    MathSciNet  MATH  Google Scholar 

  14. S. Brendle, Ricci Flow and the Sphere Theorem, Grad. Stud. Math., 111, Amer. Math. Soc., Providence, RI, 2010.

  15. Brendle S., Huisken G., Sinestrari C.: Ancient solutions to the Ricci flow with pinched curvature. Duke Math. J. 158, 537–551 (2011)

    Article  MATH  Google Scholar 

  16. Brendle S., Marques F.C.: Blow-up phenomena for the Yamabe equation II. J. Differential Geom. 81, 225–250 (2009)

    MathSciNet  MATH  Google Scholar 

  17. Brendle S., Schoen R.: Classification of manifolds with weakly 1/4-pinched curvatures. Acta Math. 200, 1–13 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Brendle S., Schoen R.: Manifolds with 1/4-pinched curvature are space forms. J. Amer. Math. Soc. 22, 287–307 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Brendle S., Schoen R.: Curvature, sphere theorems, and the Ricci flow. Bull. Amer. Math. Soc. (N.S.) 48, 1–32 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Chen H.: Pointwise 1/4-pinched 4-manifolds. Ann. Global Anal. Geom. 9, 161–176 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  21. Chen X.: Weak limits of Riemannian metrics in surfaces with integral curvature bounds. Calc. Var. Partial Differential Equations 6, 189–226 (1998)

    Article  MathSciNet  Google Scholar 

  22. Chen X.: Calabi flow in Riemann surface revisited: a new point of view. Internat. Math. Res. Notices 2001, 275–297 (2001)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  24. Chow B.: The Ricci flow on the 2-sphere. J. Differential Geom. 33, 325–334 (1991)

    MathSciNet  MATH  Google Scholar 

  25. Chow B.: The Yamabe flow on locally conformally flat manifolds with positive Ricci curvature. Comm. Pure Appl. Math. 45, 1003–1014 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  26. Daskalopoulos P., Hamilton R.S.: Geometric estimates for the logarithmic fast diffusion equation. Comm. Anal. Geom. 12, 143–164 (2004)

    MathSciNet  MATH  Google Scholar 

  27. DeTurck D.M.: Deforming metrics in the direction of their Ricci tensors. J. Differential Geom. 18, 157–162 (1983)

    MathSciNet  MATH  Google Scholar 

  28. Ecker K.: Heat equations in geometry and topology. Jahresber. Deutsch. Math.-Verein. 110, 117–141 (2008)

    MathSciNet  MATH  Google Scholar 

  29. Fraser A.M.: Fundamental groups of manifolds with positive isotropic curvature. Ann. of Math. (2) 158, 345–354 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  30. Hamilton R.S.: Three-manifolds with positive Ricci curvature, J. Differential Geom. 17, 255–306 (1982)

    MathSciNet  MATH  Google Scholar 

  31. Hamilton R.S.: Four-manifolds with positive curvature operator. J. Differential Geom. 24, 153–179 (1986)

    MathSciNet  MATH  Google Scholar 

  32. Hamilton R.S.: The Ricci flow on surfaces. Contemp. Math. 71, 237–262 (1988)

    MathSciNet  Google Scholar 

  33. R.S. Hamilton, Lectures on geometric flows, unpublished manuscript, 1989.

  34. Hamilton R.S.: The Harnack estimate for the Ricci flow. J. Differential Geom. 37, 225–243 (1993)

    MathSciNet  MATH  Google Scholar 

  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.

  36. Hamilton R.S.: Four-manifolds with positive isotropic curvature. Comm. Anal. Geom. 5, 1–92 (1997)

    MathSciNet  MATH  Google Scholar 

  37. Huisken G.: Ricci deformation of the metric on a Riemannian manifold. J. Differential Geom. 21, 47–62 (1985)

    MathSciNet  MATH  Google Scholar 

  38. Klingenberg W.: Über Riemannsche Mannigfaltigkeiten mit positiver Krümmung. Comment. Math. Helv. 35, 47–54 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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.

  41. Margerin C.: A sharp characterization of the smooth 4-sphere in curvature terms. Comm. Anal. Geom. 6, 21–65 (1998)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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.

  44. G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv:math/0211159.

  45. G. Perelman, Ricci flow with surgery on three-manifolds, arXiv:math/0303109.

  46. G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, arXiv:math/0307245.

  47. Schoen R.: Conformal deformation of a Riemannian metric to constant scalar curvature. J. Differential Geom. 20, 479–495 (1984)

    MathSciNet  MATH  Google Scholar 

  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.

  49. Schoen R., Yau S.-T.: On the proof of the positive mass conjecture in general relativity. Comm. Math. Phys. 65, 45–76 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  50. Schoen R., Yau S.-T.: Conformally flat manifolds, Kleinian groups. and scalar curvature. Invent. Math. 92, 47–71 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  51. Schwetlick H., Struwe M.: Convergence of the Yamabe flow for large energies. J. Reine Angew. Math. 562, 59–100 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  53. Struwe M.: A global compactness result for elliptic boundary value problems involving limiting nonlinearities. Math. Z. 187, 511–517 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  54. Struwe M.: Curvature flows on surfaces. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1, 247–274 (2002)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  56. J. Wolf, Spaces of Constant Curvature. Fifth ed., Publish or Perish, Houston, TX, 1984.

  57. Yamabe H.: On a deformation of Riemannian structures on compact manifolds. Osaka Math J. 12, 21–37 (1960)

    MathSciNet  MATH  Google Scholar 

  58. Ye R.: Global existence and convergence of Yamabe flow. J. Differential Geom. 39, 35–50 (1994)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Stanford University, Stanford, CA, 94305, USA

    Simon Brendle

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  1. Simon Brendle
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Correspondence to Simon Brendle.

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Communicated by: Hiraku Nakajima

This article is based on the 9th Takagi Lectures that the author delivered at Research Institute for Mathematical Sciences, Kyoto University on June 4, 2011.

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Brendle, S. Evolution equations in Riemannian geometry. Jpn. J. Math. 6, 45–61 (2011). https://doi.org/10.1007/s11537-011-1115-1

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  • DOI: https://doi.org/10.1007/s11537-011-1115-1

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