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Conformal entropy rigidity through Yamabe flows

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We introduce two versions of the Yamabe flow which preserve negative scalar-curvature bounds. First we show existence and smooth convergence of solutions to these flows. We then show that a metric with negative scalar curvature is controlled by the Yamabe metrics in the same conformal class with constant extremal scalar curvatures. This implies that the volume entropy of our original metric is controlled by the entropies of these Yamabe metrics. We eventually use these Yamabe flows to prove an entropy-rigidity result: when the Yamabe metric has negative sectional curvature, the entropy of a metric in the same conformal class is extremal if and only if the metric has constant extremal scalar curvature.

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References

  1. Aubin T.: Nonlinear analysis on manifolds Monge-Ampère equations. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 252. Springer, New York (1982)

    Google Scholar 

  2. Besson G., Courtois G., Gallot S.: Entropie et rigidités des espaces localement symétriques de courbure strictement négative. Geom. Funct. Anal. 5(5), 731–799 (1996)

    Article  MathSciNet  Google Scholar 

  3. Besson G., Courtois G., Gallot S.: Minimal entropy and Mostow’s rigidity theorems. Ergod. Theory Dyn. Syst. 16(4), 623–649 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  4. Besse, A.: Einstein manifolds. In: Classics in Mathematics. Springer, Berlin (2008) (reprint of the 1987 edition)

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Chow, B., Bennett, P., Lu, Ni L.: Hamilton’s Ricci flow. In: Graduate Studies in Mathematics, vol. 77 (2008)

  8. Connell C., Farb B.: Minimal entropy rigidity for lattices in products of rank one symmetric spaces. Commun. Anal. Geom. 11(5), 1001–1026 (2003)

    MathSciNet  MATH  Google Scholar 

  9. Freire A., Mañé R.: On the entropy of the geodesic flow in manifolds without conjugate points. Invent Math. 69, 375–392 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gromov M., Thurston W.: Pinching constants for hyperbolic manifolds. Invent. Math. 89(1), 1–12 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hamilton, R.: Lectures on geometric flows (1989, unpublished manuscript)

  12. Hamenstädt U.: Entropy-rigidity of locally symmetric spaces of negative curvature. Ann. Math. (2) 131(1), 35–51 (1990)

    Article  MATH  Google Scholar 

  13. Jane D.: An example of how the Ricci flow can increase topological entropy. Ergod. Theory Dyn. Syst. 27(6), 1919–1932 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Katok, A., Hasselblatt, B.: Introduction to the modern theory of dynamical systems. In: Encyclopedia of Mathematics and its Applications (with a supplementary chapter by Katok and Leonardo Mendoza), vol. 54. Cambridge University Press, Cambridge (1995).

  15. Katok A., Knieper G., Weiss H.: Formulas for the derivative and critical points of topological entropy for Anosov and geodesic flows. Commun. Math. Phys. 138(1), 19–31 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  16. Knieper G.: The uniqueness of the maximal measure for geodesic flows on symmetric spaces of higher rank. Israel J. Math. 149, 171–183 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kobayashi O.: Scalar curvature of a metric with unit volume. Math. Ann. 279, 253–265 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ledrappier F., Wang X.: An integral formula for the volume entropy with applications to rigidity. J. Differ. Geom. 85(3), 461–478 (2010)

    MathSciNet  MATH  Google Scholar 

  19. Lieberman G.: Second Order Parabolic Differential Equations. World Scientific Publishing, River Edge (1996)

    MATH  Google Scholar 

  20. Manning A.: Topological entropy for geodesic flows. Ann. Math. (2) 110(3), 567–573 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  21. Manning A.: The volume entropy of a surface decreases along the Ricci flow. Ergod. Theory Dyn. Syst. 24(1), 171–176 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mostow G., Siu Y.T.: A compact Kähler surface of negative curvature not covered by the ball. Ann. Math. (2) 112(2), 321–360 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  23. Robert, G.: Invariants topologiques et géométriques reliés aux longueurs des géodésiques et aux sections harmoniques de fibrés. PhD thesis, Institut Fourier de Grenoble

  24. 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 

  25. Trudinger N.: 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 

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

    MathSciNet  MATH  Google Scholar 

  27. Yau S.T.: Remarks on conformal transformations. J. Differ. Geom 8, 369–381 (1973)

    MATH  Google Scholar 

  28. Ye R.: Global existence and convergence of Yamabe flow. J. Differ. Geom. 39(1), 35–50 (1994)

    MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. Instituto de Matemáticas, Universidad Nacional Autonóma de México, Ciudad Universitaria, Coyoacán, 04510, Mexico, D.F., Mexico

    Pablo Suárez-Serrato

  2. Departament de Matemàtica Aplicada 1, Universitat Politècnica de Catalunya, Barcelona, Spain

    Pablo Suárez-Serrato

  3. Laboratoire Jean Leray, Université de Nantes 2, rue de la Houssinière, BP 92208, 44322, Nantes Cedex 3, France

    Samuel Tapie

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  1. Pablo Suárez-Serrato
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  2. Samuel Tapie
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Correspondence to Pablo Suárez-Serrato.

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Suárez-Serrato, P., Tapie, S. Conformal entropy rigidity through Yamabe flows. Math. Ann. 353, 333–357 (2012). https://doi.org/10.1007/s00208-011-0687-7

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  • DOI: https://doi.org/10.1007/s00208-011-0687-7

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