Advertisement

First eigenvalue monotonicity for the p-Laplace operator under the Ricci flow

  • Jia Yong WuEmail author
Article

Abstract

In this note, we discuss the monotonicity of the first eigenvalue of the p-Laplace operator (p ≥ 2) along the Ricci flow on closed Riemannian manifolds. We prove that the first eigenvalue of the p-Laplace operator is nondecreasing along the Ricci flow under some different curvature assumptions, and therefore extend some parts of Ma’s results [Ann. Glob. Anal. Geom., 29, 287–292 (2006)].

Keywords

Ricci flow first eigenvalue p-Laplace operator monotonicity 

MR(2000) Subject Classification

58C40 53C44 

References

  1. [1]
    Hamilton, R. S.: Three manifolds with positive Ricci curvature. J. Diff. Geom., 17(2), 255–306 (1982)MathSciNetzbMATHGoogle Scholar
  2. [2]
    Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv: math.DG/0211159v1 (2002)Google Scholar
  3. [3]
    Cao, X. D.: Eigenvalues of (−Δ + R/2) on manifolds with nonnegative curvature operator. Math. Ann., 337(2), 435–441 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    Li, J. F.: Eigenvalues and energy functionals with monotonicity formulae under Ricci flow. Math. Ann., 338(4), 927–946 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    Cao, X. D.: First eigenvalues of geometric operators under the Ricci flow. Proc. Amer. Math. Soc., 136, 4075–4078 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    Ma, L.: Eigenvalue monotonicity for the Ricci-Hamilton flow. Ann. Glob. Anal. Geom., 29, 287–292 (2006)zbMATHCrossRefGoogle Scholar
  7. [7]
    Grosjean, J. F.: p-Laplace operator and diameter of manifolds. Ann. Glob. Anal. Geom., 28, 257–270 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    Kawai, S., Nakauchi, N.: The first eigenvalue of the p-Laplacian on a compact Riemannian manifold. Nonlin. Anal., 55, 33–46 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    Kotschwar, B., Ni, L.: Local gradient estimates of p-harmonic functions, 1/H-flow, and an entropy formula. Ann. Sci. Ec. Norm. Sup., 42(1), 1–36 (2009)MathSciNetzbMATHGoogle Scholar
  10. [10]
    Ma, L., Chen, D. Z., Yang, Y.: Some results on subelliptic equations. Acta Mathematica Sinica, English Series, 22(6), 1965–1704 (2006)MathSciNetCrossRefGoogle Scholar
  11. [11]
    Matei, A. M.: First eigenvalue for the p-Laplace operator. Nonlinear Anal., 39, 1051–1068 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    Serrin, J.: Local behavior of solutions of quasi-linear equations. Acta Math., 111, 247–302 (1964)MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    Tolksdorff, P.: Regularity for a more general class of quasilinear ellptic equations. J. Diff. Equa., 51, 126–150 (1984)CrossRefGoogle Scholar
  14. [14]
    Kato, T.: Perturbation Theory for Linear Operator, 2nd edtion, Springer, Berlin, Heidelberg, New York, Tokyo, 1984Google Scholar
  15. [15]
    Kleiner, B., Lott, J.: Note on Perelman’s papers. Geometry and Topology, 12, 2587–2855 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press Harcourt Brace Jovanovich Publishers, New York, 1978zbMATHGoogle Scholar
  17. [17]
    Chow, B., Lu, P., Ni, L.: Hamilton’s Ricci Flow. Lectures in Contemporary Mathematics 3, American Mathematical Society, Providence, RI, 2006zbMATHGoogle Scholar
  18. [18]
    Ling, J.: A comparison theorem and a sharp bound via the Ricci flow. arXiv: math.DG/0710.2574v1 (2007)Google Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Department of MathematicsShanghai Maritime UniversityShanghaiP. R. China

Personalised recommendations