The Journal of Geometric Analysis

, Volume 29, Issue 2, pp 1116–1135 | Cite as

Monotonicity of Eigenvalues and Functionals Along the Ricci–Bourguignon Flow

  • Lin Feng WangEmail author


In this paper we first prove the monotonicity of the lowest eigenvalue of the Schrödinger operator
$$\begin{aligned} \frac{(1-(n-1)\rho )^2}{1-2(n-1)\rho }R-4\varDelta \end{aligned}$$
along the Ricci–Bourguignon flow
$$\begin{aligned} \frac{\partial \mathrm {g}}{\partial t}=-2(\mathrm {Ric}-\rho R\mathrm {g}) \end{aligned}$$
based on an evolving formula of the \(\mathcal {F}_{\rho }\)-functional, and rule out nontrivial steady breathers. Then we prove the monotonicity of the lowest eigenvalue of the Schrödinger operator \(BR-4\varDelta \) along the Ricci–Bourguignon flow for any constant B satisfying
$$\begin{aligned} B\ge \frac{4(1-(n-1)\rho )^2-n\rho }{4(1-2(n-1)\rho )}>0 \end{aligned}$$
for the case that \(\rho \le 0.\) We also study the evolving formula of the \(\mathcal {W}_{\rho }\)-functional and get the monotonicity of the infimum of the \(\mathcal {W}_{\rho }\)-functional, based on which we can prove that a shrinking breather should be an Einstein metric for the case that \(\rho <0.\)


Ricci–Bourguignon flow Eigenvalue \(\mathcal {F}_{\rho }\)-functional \(\mathcal {W}_{\rho }\)-functional Evolving formula Breather 

Mathematics Subject Classification



  1. 1.
    Brendle, S.: Convergence of the Yamabe flow for arbitrary initial energy. J. Differ. Geom. 69(2), 217–278 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Cao, X.D.: Eigenvalues of \(-\triangle +\frac{1}{2}R\) on manifolds with nonnegative curvature operator. Math. Ann. 337(2), 435–441 (2007)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Cao, X.D.: First eigenvalues of geometric operators under the Ricci flow. Proc. Am. Math. Soc. 136(11), 4075–4078 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cao, X.D.: Differential Harnack estimates for backward heat equations with potentials under the Ricci flow. J. Funct. Anal. 255(4), 1024–1038 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cao, X.D., Hou, S.B., Ling, J.: Estimate and monotonicity of the first eigenvalue under the Ricci flow. Math. Ann. 354(2), 451–463 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Catino, G., Cremaschi, L., Djadli, Z., Mantegazza, C., Mazzieri, L.: The Ricci–Bourguignon flow. Pac. J. Math. 287(2), 337–370 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chow, B., Chu, S.-C., Glickenstein, D., Guenther, C., Isenberg, J., Ivey, T., Knopf, D., Lu, P., Luo, F., Ni, L.: The Ricci Flow: Techniques and Applications: Part II: Analytic Aspects. Mathematical Surveys and Monographs, vol. 144. American Mathematical Society (2008)Google Scholar
  8. 8.
    Fang, S.W., Xu, H.F., Zhu, P.: Evolution and monotonicity of eigenvalues under the Ricci flow. Sci. China Math. 58(8), 1–8 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hamilton, R.S.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17(2), 255–306 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hamilton, R.S.: The Harnack estimate for the Ricci flow. J. Differ. Geom. 37(1), 225–243 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kleiner, B., Lott, J.: Notes on Perelman’s papers. Geom. Topol. 12(5), 2587–2855 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Li, J.-F.: Eigenvalues and energy functionals with monotonicity formulae under Ricci flow. Math. Ann. 338(4), 927–946 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Müller, R.: Differential Harnack inequalities and the Ricci flow. EMS Series of Lectures in Mathematics. European Mathematical Society, Zurich (2006)Google Scholar
  14. 14.
    Müller, R.: Ricci flow coupled with harmonic map heat flow. PhD Thesis, ETH Zurich, No. 18290 (2009)Google Scholar
  15. 15.
    Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159 (2002)
  16. 16.
    Pigola, S., Rigoli, M., Rimoldi, M.: Ricci almost solitons. Ann. Della Scuola Normale Superiore Di Pisa-Classe Di Sci. 10(4), 757–799 (2010)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Rothaus, O.S.: Logarithmic Sobolev inequalities and the spectrum of Schrödinger operators. J. Funct. Anal. 42(1), 110–120 (1981)CrossRefzbMATHGoogle Scholar
  18. 18.
    Wang, L.F.: Differential Harnack inequalities under a coupled Ricci flow. Math. Phys. Anal. Geom. 15(4), 343–360 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Wang, E.-M., Zheng, Y.: Regularity of the first eigenvalue of the \(p\)-Laplacian and Yamabe invariant along geometric flows. Pac. J. Math. 254(1), 239–255 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Wu, J.Y., Wang, E.-M., Zheng, Y.: First eigenvalue of the \(p\)-Laplace operator along the Ricci flow. Ann. Glob. Anal. Geom. 38(1), 27–55 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Ye, R.: Global existence and convergence of Yamabe flow. J. Differ. Geom. 39(1), 35–50 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Zeng, F., He, Q., Chen, B.: Monotonicity of eigenvalues of geometric operators along the Ricci–Bourguignon flow. arXiv:math.DG/1512.08158v1 (2015)

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© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.School of ScienceNantong UniversityNantongChina

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