Skip to main content
SpringerLink
Log in

The Differentiable Sphere Theorem (After S. Brendle and R. Schoen)

  • Chapter
  • First Online:
Ricci Flow and Geometric Applications

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2166))

  • 1808 Accesses

Abstract

In these notes we describe a major result obtained recently using the Ricci flow technique in the context of positive curvature. It is due to S. Brendle and R. Schoen and states that a strictly 1/4-pinched closed manifold carries a metric of constant (positive) sectional curvature. It relies on a technique developed by C. Böhm and B. Wilking who obtained the same conclusion assuming that the manifold has positive curvature operator. The maximum principle applied to the Ricci flow equation leads to studying an ordinary differential equation on the space of curvature operators.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 49.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

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

    Google Scholar 

  2. M. Berger, Sur quelques variétés riemanniennes suffisamment pincées. Bull. Soc. Math. France 88, 57–71 (1960)

    MathSciNet  MATH  Google Scholar 

  3. M. Berger, Sur les variétés riemanniennes pincées juste au-dessous de 1∕4. Ann. Inst. Fourier (Grenoble) 33 (2), 135–150 (loose errata) (1983)

    Google Scholar 

  4. M. Berger, A Panoramic View of Riemannian Geometry (Springer, Berlin, 2003)

    Book  MATH  Google Scholar 

  5. G. Besson, Preuve de la conjecture de Poincaré en déformant la métrique par la courbure de Ricci (d’après Perelman), in Séminaire Bourbaki 2004–2005. Astérisque, vol. 307 (Société mathématiques de France, Paris, 2006), pp. 309–348

    Google Scholar 

  6. C. Böhm, B. Wilking, Manifolds with positive curvature operators are space forms. Ann. Math. (2) 167 (3), 1079–1097 (2008)

    Google Scholar 

  7. S. Brendle, A general convergence result for the Ricci flow in higher dimensions. Duke Math. J. 145 (3), 585–601 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. S. Brendle, Einstein manifolds with nonnegative isotropic curvature are locally symmetric. Duke Math. J. 151 (1), 1–21 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. S. Brendle, R. Schoen, Manifolds with 1∕4-pinched curvature are space forms. J. Am. Math. Soc. 22 (1), 287–307 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. S. Brendle, R. Schoen, Sphere theorems in geometry, in Handbook of Geometric Analysis, No. 3, Advanced Lectures in Mathematics (ALM), vol. 14 (International Press, Somerville, MA, 2010), pp. 41–75

    Google Scholar 

  12. H. Chen, Pointwise \(\frac{1} {4}\)-pinched 4-manifolds. Ann. Global Anal. Geom. 9 (2), 161–176 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  13. B.-L. Chen, X.-P. Zhu, Ricci flow with surgery on four-manifolds with positive isotropic curvature. J. Differ. Geom. 74 (2), 177–264 (2006)

    MathSciNet  MATH  Google Scholar 

  14. B. Chow, D. Knopf, The Ricci Flow: An Introduction. Mathematical Surveys and Monographs, vol. 110 (American Mathematical Society, Providence, RI, 2004)

    Google Scholar 

  15. B. Chow, S.-C. Chu, D. Glickenstein, C. Guenther, J. Isenberg, T. Ivey, D. Knopf, P. Lu, F. Luo, L. Ni, The Ricci Flow: Techniques and Applications. Part I. Mathematical Surveys and Monographs, vol. 135 (American Mathematical Society, Providence, RI, 2007). Geometric Aspects

    Google Scholar 

  16. B. Chow, S.-C. Chu, D. Glickenstein, C. Guenther, J. Isenberg, T. Ivey, D. Knopf, P. Lu, F. Luo, L. Ni, The Ricci Flow: Techniques and Applications. Part II. Mathematical Surveys and Monographs, vol. 144 (American Mathematical Society, Providence, RI, 2008). Analytic Aspects

    Google Scholar 

  17. A. Fraser, Fundamental groups of manifolds with positive isotropic curvature. Ann. Math. (2) 158 (1), 345–354 (2003)

    Google Scholar 

  18. A. Fraser, J. Wolfson, The fundamental group of manifolds of positive isotropic curvature and surface groups. Duke Math. J. 133 (2), 325–334 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. S. Gadgil, H. Seshadri, On the topology of manifolds with positive isotropic curvature. Proc. Am. Math. Soc. 137 (5), 1807–1811 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry, Universitext, 3rd edn. (Springer, Berlin, 2004)

    Book  MATH  Google Scholar 

  21. R. Hamilton, Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17, 255–306 (1982)

    MathSciNet  MATH  Google Scholar 

  22. R. Hamilton, Four-manifolds with positive curvature operator. J. Differ. Geom. 24, 153–179 (1986)

    MathSciNet  MATH  Google Scholar 

  23. R. Hamilton, A compactness property for solutions of the Ricci flow. Am. J. Math. 117 (3), 545–572 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  24. R. Hamilton, The formation of singularities in the Ricci flow, in Surveys in Differential Geometry, vol. II (International Press, Cambridge, MA, 1995), pp. 7–136

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. L. Hernández, Kähler manifolds and 1∕4-pinching. Duke Math. J. 62 (3), 601–611 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  27. G. Huisken, Ricci deformation of the metric on a Riemannian manifold. J. Diff. Geom. 21 (1), 47–62 (1985)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  29. C. Margerin, A sharp characterization of the smooth 4-sphere in curvature terms. Commun. Anal. Geom. 6 (1), 21–65 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  30. M. Micallef, J. Moore, Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes. Ann. Math. 127, 199–227 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  31. M. Micallef, McK. Wang, Metrics with nonnegative isotropic curvature. Duke Math. J. 72 (3), 649–672 (1993)

    Google Scholar 

  32. H. Nguyen, Isotropic curvature and the Ricci flow. Int. Math. Res. Not. IMRN 2010 (3), 536–558 (2010)

    MathSciNet  MATH  Google Scholar 

  33. S. Nishikawa, Deformation of Riemannian metrics and manifolds with bounded curvature ratios, in Geometric Measure Theory and the Calculus of Variations (Arcata, Calif., 1984). Proceedings of Symposia in Pure Mathematics, vol. 44 (American Mathematical Society, Providence, RI, 1986)

    Google Scholar 

  34. G. Perelman, The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159, November 2002

    Google Scholar 

  35. G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. arXiv:math.DG/0307245, July 2003

    Google Scholar 

  36. G. Perelman, Ricci flow with surgery on three-manifolds. arXiv:math.DG/0303109, March 2003

    Google Scholar 

  37. P. Petersen, T. Tao, Classification of almost quarter-pinched manifolds. Proc. Am. Math. Soc. 137 (7), 2437–2440 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  38. H.E. Rauch, A contribution to differential geometry in the large. Ann. Math. (2) 54, 38–55 (1951)

    Google Scholar 

  39. H. Seshadri, Manifolds with nonnegative isotropic curvature. Commun. Anal. Geom. 17 (4), 621–635 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  40. S.-T. Yau, F. Zheng, Negatively \(\frac{1} {4}\)-pinched Riemannian metric on a compact Kähler manifold. Invent. Math. 103 (3), 527–535 (1991)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author is supported by ERC Advanced Grant 320939, “Geometry and Topology of Open Manifolds” (GETOM).

Author information

Authors and Affiliations

  1. Université Grenoble Alpes – Institut Fourier, CS40700, 38058, Grenoble Cedex 9, France

    Gérard Besson

Authors
  1. Gérard Besson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gérard Besson .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Pisa, Pisa, Italy

    Riccardo Benedetti

  2. Department of Mathematics, University of Naples, Naples, Italy

    Carlo Mantegazza

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Besson, G. (2016). The Differentiable Sphere Theorem (After S. Brendle and R. Schoen). In: Benedetti, R., Mantegazza, C. (eds) Ricci Flow and Geometric Applications. Lecture Notes in Mathematics(), vol 2166. Springer, Cham. https://doi.org/10.1007/978-3-319-42351-7_1

Download citation

Publish with us

Policies and ethics

Search

Navigation