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Yamabe flow and metrics of constant scalar curvature on a complete manifold

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Abstract

In this paper, we give a condition on the initial metric which makes the global existence of Yamabe flow and we study the global behavior of the Yamabe flow in a complete noncompact Riemannian manifold. As application we study the ADM mass monotonicity of Yamabe flow in AF manifolds. We use the variational method to study the existence problem of metrics with constant scalar curvature on complete non-compact Riemannian manifolds and we can give a partial affirmative answer to a question posed by Kazdan (Math Ann 261(2):227–234, 1982).

Mathematics Subject Classification

53C20 35Jxx 58J20 

Notes

Acknowledgements

The author is very grateful to the unknown referees for helpful suggestions, in particular for pointing out the reference [32]. Part of the revision had been done when the author visited the Department of Mathematics at Stanford University in June and July of 2018 and the author would like to thank Prof. R. Schoen for the invitation.

References

  1. 1.
    An, Y., Ma, L.: The Maximum Principle and the Yamabe Flow, Partial Differential Equations and Their Applications, pp. 211–224. World Scientific, Singapore (1999)zbMATHGoogle Scholar
  2. 2.
    Bartnik, R.: The mass of an asymptotically flat manifold. Commun. Pure Appl. Math. 39, 661–693 (1986)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Brendle, S.: Convergence of the Yamabe flow for arbitrary initial energy. J. Differ. Geom. 69, 217–278 (2005)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Brendle, S.: Convergence of the Yamabe flow in dimension 6 and higher. Invent. Math. 170, 541–576 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cheng, L., Zhu, A.: Yamabe flow and ADM mass on asymptotically flat manifolds. J. Math. Phys. 56, 101507 (2015).  https://doi.org/10.1063/1.4934725 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chow, B.: Yamabe flow on locally conformally flat manifolds. Commun. Pure Appl. Math. XLV, 1003–1014 (1992)CrossRefGoogle Scholar
  7. 7.
    Chow, B., Lu, P., Ni, L.: Hamilton’s Ricci Flow, Graduate Studies in Mathematics, vol. 77. American Mathematical Society, Providence (2006)Google Scholar
  8. 8.
    Chrusciel, P.T., Herzlich, M.: The mass of asymptotically hyperbolic Riemannian manifolds. Pac. J. Math. 212, 231–264 (2003)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dai, X., Ma, L.: Mass under Ricci flow. Commun. Math. Phys. 274, 65–80 (2007)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Daskalopoulos, P., Sesum, N.: The classification of locally conformally flat Yamabe solitons. Adv. Math. 240, 346–369 (2013)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Ecker, K., Knopf, D., Ni, L., Topping, P.: Local monotonicity and mean value formulas for evolving Riemannian manifolds. J. Reine Angew. Math. 616, 89–130 (2008).  https://doi.org/10.1515/CRELLE.2008.019 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fischer-Colbrie, D., Schoen, R.: The structure of complete stable minimal surfaces in 3-manifolds of non-negative scalar curvature. Commun. Pure Appl. Math. 33, 199–211 (1980)CrossRefGoogle Scholar
  13. 13.
    Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs (1964)zbMATHGoogle Scholar
  14. 14.
    Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order. Springer, New York (1983)CrossRefGoogle Scholar
  15. 15.
    Hamilton, R.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 2, 255–306 (1982)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Hamilton, R.S.: Formation of singularities in the Ricci flow. Surv. Differ. Geom. 2, 7–136 (1995)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Hebey, E.: Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities. Courant Lecture Notes in Mathematics, vol. 5. New York University, Courant Institute of Mathematical Sciences, New York (1999). x+309 pp. ISBN: 0-9658703-4-0; 0-8218-2700-6Google Scholar
  18. 18.
    Kazdan, J.L.: Deformation to positive scalar curvature on complete manifolds. Math. Ann. 261(2), 227–234 (1982)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Krylov, N., Safonov, M.: A certain property of solutions of parabolic equations with measurable coefficients. Math. USSR Zzv. 16, 151–164 (1981)CrossRefGoogle Scholar
  20. 20.
    Ladyzhenskaya, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasilinear Equations of Parabolic Type, AMS Translation of Monograph, vol. 23. American Mathematical Society, Providence (1968)CrossRefGoogle Scholar
  21. 21.
    Lee, J.M., Parker, T.H.: The Yamabe problem. Bull. Am. Math. Soc. 17, 37–91 (1987)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Ma, L.: Gap theorems for locally conformally flat manifolds. J. Differ. Equ. 260(2), 1414–1429 (2016)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Ma, L.: Expanding Ricci solitons with pinched Ricci curvature. Kodai Math. J. 34, 140–143 (2011)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Ma, L., Cheng, L.: Yamabe flow and Myers type theorem on complete manifolds. J. Geom. Anal. 24(1), 246–270 (2014).  https://doi.org/10.1007/s12220-012-9336-y MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Ma, L., Cheng, L., Zhu, A.: Extending Yamabe flow on complete Riemannian manifolds. Bull. Sci. Math. 136, 882–891 (2012)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Moss, W.F., Piepenbrink, J.: Positive solutions of elliptic equation. Pac. J. Math. 75, 219–226 (1978)CrossRefGoogle Scholar
  27. 27.
    Ni, L.: The Poisson equation and Hermitian–Einstein metrics on holomorphic vector boundle over complete non-compact Kaehler manifolds. Indiana Univ. Math. J. 51(3), 679–704 (2002)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Schulz, M.: Instantaneously complete Yamabe flow on hyperbolic space. arxiv:1612.02745v1
  29. 29.
    Schwetlick, H., Struwe, M.: Convergence of the Yamabe flow for ’large’ energies. J. Reine Angew. Math. 562, 59–100 (2003)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Souplet, P., Zhang, Q.: Sharp gradient estimate and Yau’s Liouville theorem for the heat equation on noncompact manifolds. Bull. Lond. Math. Soc. 38, 1045–1053 (2006)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Schoen, R., Yau, S.T.: Conformally flat manifolds, Kleinian groups and scalar curvature. Invent. Math. 92, 47–71 (1988)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Trudinger, N.S.: Pointwise estimates and quasilinear parabolic equations. Commun. Pure Appl. Math. XXI, 205–226 (1968)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Wang, X.: The mass of asymptotically hyperbolic manifolds. J. Differ. Geom. 57(2), 273–299 (2001)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Yamabe, H.: On a deformation of Riemannian structures on compact manifolds. Osaka Math. J. 12, 21–37 (1960)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Ye, R.: Global existence and convergence of the Yamabe flow. J. Differ. Geom. 39, 35–50 (1994)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and PhysicsUniversity of Science and Technology BeijingBeijingPeople’s Republic of China
  2. 2.Department of MathematicsHenan Normal UniversityXinxiangPeople’s Republic of China

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