First eigenvalues of geometric operator under the Ricci-Bourguignon flow

Shahroud Azami


Abstract


Let $(M,g(t))$ be a compact Riemannian manifold  and  the metric $g(t)$ evolve by the Ricci-Bourguignon flow. We find the formula variation of the eigenvalues of  geometric operator $-\Delta_{\phi}+cR$ under  the Ricci-Bourguignon flow, where  $\Delta_{\phi}$  is the Witten-Laplacian operator and $R$ is the scalar curvature. In the final  we show that some quantities dependent to the eigenvalues of  the geometric operator are  nondecreasing along the Ricci-Bourguignon flow on  closed manifolds  with nonnegative curvature.

Keywords


Laplace, Ricci-Bourguignon flow

References


bibitem{JPB} J. P. Bourguignon, Ricci curvature and Einstein metrics, Global differential geometry and global analysis (Berlin,1979) Lecture nots in Math. vol. 838, Springer, Berlin, 1981, 42-63.

bibitem{XC1} X. D. Cao, Eigenvalues of $(-Delta+frac{R}{2})$ on manifolds with nonnegative curvature operator. Math. Ann. 337 (2) (2007), 435-441.

bibitem{XC2} X. D. Cao, First eigenvalues of geometric operators under the Ricci flow, Proc. Amer. Math. Soc. 136 (2008), 4075-4078.

bibitem{GC} G. Catino, L. Cremaschi, Z. Djadli, C. Mantegazza, L. Mazzieri, The Ricci-Bourguignon flow, Pacific J. Math. (2015).

bibitem{LF} L. F. D. Cerbo, Eigenvalues of the Laplacian under the Ricci flow, Rendiconti di Mathematica, Serie VII, Volume

, Roma (2007), 183-195.

bibitem{CY} Q. -M. Cheng and H. C. Yang, Estimates on eigenvalues of Laplacian, Math. Ann., 331 (2005),

-460.

bibitem{SF} S. Fang and F. Yang, First eigenvalues of geometric operators under the Yamabe flow, Bull. Korean Math. Soc. 53 (2016), 1113-1122.

bibitem{JFL} J. F. Li, Eigenvalues and energy functionals with monotonicity formula under Ricci flow, Math. Ann. (2007) 338,

-946.

bibitem{GP} G. Perelman, The entropy formula for the Ricci flow and its geometric applications (2002), ArXiv: 0211159.

bibitem{FSW} F. S. Wen, X. H. Feng, Z. Peng, Evolution and monotonicity of eigenvalues under the Ricci flow, Sci. China Math. 58 (2015),no. 8, 1737-1744.

bibitem{JY} J. Y. Wu, First eigenvalue monotonicity for the $p$-Laplace operator under the Ricci flow,

Acta mathematica Sinica, English senes, Vol. 27, NO.8 (2011), 1591-1598.

bibitem{FZ} F. Zeng, Q. He, B. Chen, Monotonicity of eigenvalues of geometric operators along the Ricci-Bourguignon flow, Arxiv, 152.08158v1.


Refbacks

  • There are currently no refbacks.


Journal of the Indonesian Mathematical Society ( p-ISSN:2086-8952 | e-ISSN:2460-0245) published by the Indonesian Mathematical Society (IndoMS).

Indexed by:

logo DOAJLogo SintaThe Indonesian Publication Index-Portal Garuda Google Scholar logo zbMath Logo AMSLogo CrossrefLogo Thomson Reuters

Visitor Number : web statistics View My Stats


Creative Commons License
Journal of the Indonesian Mathematical Society by http://jims-a.org/index.php/jimsa is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

p-ISSN:2086-8952e-ISSN:2460-0245