First eigenvalues of geometric operators under the Yamabe flow

Article
First Online:
Received:
Accepted:
  • 22 Downloads

Abstract

Suppose \((M,g_0)\) is a compact Riemannian manifold without boundary of dimension \(n\ge 3\). Using the Yamabe flow, we obtain estimate for the first nonzero eigenvalue of the Laplacian of \(g_0\) with negative scalar curvature in terms of the Yamabe metric in its conformal class. On the other hand, we prove that the first eigenvalue of some geometric operators on a compact Riemannian manifold is nondecreasing along the unnormalized Yamabe flow under suitable curvature assumption. Similar results are obtained for manifolds with boundary and for CR manifold.

Keywords

Yamabe flow Eigenvalue CR manifold 

Mathematics Subject Classification

Primary 53C44 58C40 Secondary 35K55 53C21 58J35 
This is a preview of subscription content, log in to check access.

References

  1. 1.
    Aubin, T.: Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire. J. Math. Pures Appl. (9) 55, 269–296 (1976)MathSciNetMATHGoogle Scholar
  2. 2.
    Barbosa, E., Ribeiro Jr., E.: On conformal solutions of the Yamabe flow. Arch. Math. (Basel) 101, 79–89 (2013)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Brendle, S.: A generalization of the Yamabe flow for manifolds with boundary. Asian J. Math. 6, 625–644 (2002)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Brendle, S.: Convergence of the Yamabe flow for arbitrary initial energy. J. Differ. Geom. 69, 217–278 (2005)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Brendle, S.: Convergence of the Yamabe flow in dimension 6 and higher. Invent. Math. 170, 541–576 (2007)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Brendle, S., Chen, S.-Y.S.: An existence theorem for the Yamabe problem on manifolds with boundary. J. Eur. Math. Soc. (JEMS) 16, 991–1016 (2014)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Cao, X.: Eigenvalues of \((-\Delta +\frac{R}{2})\) on manifolds with nonnegative curvature operator. Math. Ann. 377, 435–441 (2007)MathSciNetGoogle Scholar
  8. 8.
    Cao, X.: First eigenvalues of geometric operators under the Ricci flow. Proc. Am. Math. Soc. 136, 4075–4078 (2008)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Cao, X., Hou, S., Ling, J.: Estimate and monotonicity of the first eigenvalue under the Ricci flow. Math. Ann. 354, 451–463 (2012)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Chang, S.C., Cheng, J.H.: The Harnack estimate for the Yamabe flow on CR manifolds of dimension 3. Ann. Glob. Anal. Geom. 21, 111–121 (2002)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Chang, S.C., Chiu, H.L., Cheng, J.H.: The Li–Yau–Hamilton inequality for Yamabe flow on a closed CR \(3\)-manifold. Trans. Am. Math. Soc. 362, 1681–1698 (2010)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Chang, S.C., Lin, C.Y., Wu, C.T.: Eigenvalues and energy functional with monotonicity formulae under the CR Yamabe flow on a closed pseudohermitian 3-manifold. Nonlinear Stud. 18, 377–392 (2011)MathSciNetMATHGoogle Scholar
  13. 13.
    Cheng, J.H., Chiu, H.L., Yang, P.: Uniformization of spherical CR manifolds. Adv. Math. 255, 182–216 (2014)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Cheng, J.H., Malchiodi, A., Yang, P.: A positive mass theorem in three dimensional Cauchy–Riemann geometry. Adv. Math. 308, 276–347 (2017)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Chow, B.: The Yamabe flow on locally conformally flat manifolds with positive Ricci curvature. Commun. Pure Appl. Math. 45, 1003–1014 (1992)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Daskalopoulos, P., del Pino, M., King, J., Sesum, N.: Type I ancient compact solutions of the Yamabe flow. Nonlinear Anal. 137, 338–356 (2016)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Escobar, J.F.: Uniqueness and non-uniqueness of metrics with prescribed scalar and mean curvature on compact manifolds with boundary. J. Funct. Anal. 202, 424–442 (2003)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Escobar, J.F.: The Yamabe problem on manifolds with boundary. J. Differ. Geom. 35, 21–84 (1992)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Gamara, N.: The CR Yamabe conjecture—the case \(n=1\). J. Eur. Math. Soc. 3, 105–137 (2001)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Gamara, N., Yacoub, R.: CR Yamabe conjecture–the conformally flat case. Pacific J. Math. 201, 121–175 (2001)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Hamilton, R.S.: Lectures on Geometric Flows (1989), unpublishedGoogle Scholar
  22. 22.
    Han, Z.C., Li, Y.Y.: The Yamabe problem on manifolds with boundary: existence and compactness results. Duke Math. J. 99, 489–542 (1999)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Hagood, J.W., Thomson, B.S.: Recovering a function from a Dini derivative. Am. Math. Mon. 113, 34–46 (2006)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Ho, P.T.: Result related to prescribing pseudo-Hermitian scalar curvature. Int. J. Math 24, 29 (2013)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Ho, P.T.: The long time existence and convergence of the CR Yamabe flow. Commun. Contemp. Math 14, 50 (2012)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Ho, P.T.: The Webster scalar curvature flow on CR sphere. Part I. Adv. Math. 268, 758–835 (2015)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Ho, P.T.: The Webster scalar curvature flow on CR sphere. Part II. Adv. Math. 268, 836–905 (2015)MathSciNetMATHGoogle Scholar
  28. 28.
    Ho, P. T.: First eigenvalues of geometric operators under the Yamabe flow (2018). arXiv:1803.07787
  29. 29.
    Jerison, D., Lee, J.M.: Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem. J. Am. Math. Soc. 1, 1–13 (1988)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Jerison, D., Lee, J.M.: Intrinsic CR normal coordinates and the CR Yamabe problem. J. Differ. Geom. 29, 303–343 (1989)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Jerison, D., Lee, J.M.: The Yamabe problem on CR manifolds. J. Differ. Geom. 25, 167–197 (1987)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Kato, T.: Perturbation theory for linear operator, 2nd edn. Springer, Berlin (1984)Google Scholar
  33. 33.
    Kazdan, J.L., Warner, F.W.: Scalar curvature and conformal deformation of Riemannian structure. J. Differ. Geom. 10, 113–134 (1975)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Kleiner, B., Lott, J.: Note on Perelman’s papers. Geom. Topol. 12, 2587–2855 (2008)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Lee, J.M., Parker, T.H.: The Yamabe problem. Bull. Am. Math. Soc. (N.S.) 17, 37–91 (1987)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Li, J.F.: Eigenvalues and energy functionals with monotonicity formulae under Ricci flow. Math. Ann. 338, 927–946 (2007)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Li, J.F.: Evolution of eigenvalues along rescaled Ricci flow. Can. Math. Bull. 56, 127–135 (2013)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Li, P., Yau, S.T.: Estimates of eigenvalues of a compact Riemannian manifold. In: Geometry of the Laplace Operator (Honolulu 1979). Proceedings of Symposium Pure Mathematics 36, American Mathematical Society, Providence, pp. 205–239 (1980)Google Scholar
  39. 39.
    Ling, J.: The first Dirichlet eigenvalue of a compact manifold and the Yang conjecture. Math. Nachr. 280, 1354–1362 (2007)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Lou, Y.: Uniqueness and non-uniqueness of metrics with prescribed scalar curvature on compact manifolds. Indiana Univ. Math. J. 47, 1065–1081 (1998)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Ma, L.: Eigenvalue monotonicity for the Ricci-Hamilton flow. Ann. Glob. Anal. Geom. 29, 287–292 (2006)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Ma, L., Cheng, L.: Yamabe flow and Myers type theorem on complete manifolds. J. Geom. Anal. 24, 246–270 (2014)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Ma, L., Liang, L., Zhu, A.: Extending Yamabe flow on complete Riemannian manifolds. Bull. Sci. Math. 136, 882–891 (2012)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv:math/0211159
  45. 45.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics, IV, Analysis of Operators. Academic Press, Cambridge (1972)MATHGoogle Scholar
  46. 46.
    Schoen, R.: Conformal deformation of a Riemannian metric to constant scalar curvature. J. Differ. Geom. 20, 479–495 (1984)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Schwetlick, H., Struwe, M.: Convergence of the Yamabe flow for “large” energies. J. Reine Angew. Math. 562, 59–100 (2003)MathSciNetMATHGoogle Scholar
  48. 48.
    Suàrez-Serrato, P., Tapie, S.: Conformal entropy rigidity through Yamabe flows. Math. Ann. 353, 333–357 (2012)MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    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
  50. 50.
    Wu, J.Y., Wang, E.M., Zheng, Y.: First eigenvalue of the \(p\)-Laplace operator along the Ricci flow. Ann. Glob. Anal. Geom. 38, 27–55 (2010)MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Yamabe, H.: On a deformation of Riemannian structures on compact manifolds. Osaka Math. J. 12, 21–37 (1960)MathSciNetMATHGoogle Scholar
  52. 52.
    Ye, R.: Global existence and convergence of Yamabe flow. J. Differ. Geom. 39, 35–50 (1994)MathSciNetCrossRefMATHGoogle Scholar
  53. 53.
    Zhang, Y.: The contact Yamabe flow. Ph.D. thesis, University of Hanover (2006)Google Scholar
  54. 54.
    Zhao, L.: The first eigenvalue of Laplace operator under powers of mean curvature flow. Sci. China Math. 53, 1703–1710 (2010)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsSogang UniversitySeoulKorea

Personalised recommendations