The first eigenvalue of $\left(p,q\right)$-elliptic quasilinear system along the Ricci flow

Document Type : Research Article


1 Department of pure mathematics, Faculty of mathematical science, Imam Khomeini international university, Qazvin, Iran

2 Department of pure mathematics, Faculty of science, Imam Khomeini international university, Qazvin, Iran


In this paper we investigate the monotonicity of the first eigenvalue of $\left(p,q\right)$-elliptic quasilinear systems along the Ricci flow in both normalized and unnormalized conditions. In particular, we study the eigenvalue problem for this system in the case of Bianchi classes for $3$-homogeneous manifolds.


Main Subjects

  1. [1] A. Abolarinwa, Evolution and monotonicity of the first eigenvalue of p-Laplacian under the Ricci harmonic flow, J.

    Appl. Anal. 21 (2015), no. 2, 147--160.

    [2] S. Azami, Eigenvalues of a (p; q)-Laplacian system under the mean curvature flow, Iran. J. Sci. Technol. Trans. A

    Sci. 43 (2019), no. 4, 1881--1888.

    [3] K. Brown and Y. Zhang, On a system of reaction-diffusion equations describing a population with two age groups,

    1. Math. Anal. Appl. 282 (2003), no. 2, 444--452.

    [4] X. Cao, Eigenvalues of (-Δ + R/2 ) on manifolds with nonnegative curvature operator, Math. Ann. 337 (2007), no.

    2, 435--441.

    [5] X. Cao, J. Guckenheimer and L. Saloff-Coste, The backward behavior of the Ricci and cross curvature flows on

    S(2, R), Comm. Anal. Geom. 17 (2009), no. 4, 777--796.

    [6] X. Cao and L. Saloff-Coste, Backward Ricci flow on locally homogeneous 3-manifolds, Comm. Anal. Geom. 17

    (2009), no. 2, 305--325.

    [7] S. Cheng, Eigenfunctions and eigenvalues of Laplacian, Proc. Sympos. Pure Math. 27 part 2, Differential Geometry,

    (eds. S. Chern and R. Osserman), 185--193, Amer. Math. Soc., Providence, R.I., 1975.

    [8] Q. Cheng and H. Yang, Estimates on eigenvalues of Laplacian, Math. Ann. 331 (2005), no. 2, 445--460.

    [9] Y. Chen, Y. Giga and S. Goto, Uniqueness ans existence of viscosity solutions of generalized mean curvature flow

    equations, J. Differential Geom. 33 (1991) 749--786.

    [10] Y. Choi, Z. Huan and R. Lui, Global existence of solutions of a strongly coupled quasilinear parabolic system with

    application to electrochemistry, J. Differential Equations 194 (2003), no. 2, 406--432.

    [11] B. Chow and D. Knopf, The Ricci Flow: An Introduction, American Mathematical Society, Providence, RI, 2004.

    [12] A. Constantin, J. Escher and Z. Yin, Global solutions for quasilinear parabolic systems, J. Differential Equations

    197 (2004), no. 1, 73--84.

    [13] E. Dancer and Y. Du, Effects of certain degeneracies in the predator-prey model, SIAM J. Math. Anal. 34 (2002),

    1. 2, 292--314.

    [14] D. De Turk, Deforming metrics in the direction of their Ricci tensors, J. Differential Geom. 18 (1983), no. 1,


    [15] J. Eells and J. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109--160.

    [16] D. Friedan, Nonlinear models in 2 + ϵ dimensions, Ann. Physics 163 (1985), no. 2, 318--419.

    [17] R. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geometry 17 (1982), no. 2, 255--306.

    [18] S. Hou, Eigenvalues under the backward Ricci flow on locally homogeneous closed 3-manifolds, Acta Math. Sin.

    (Engl. Ser.) 34 (2018), no. 7, 1179--1194.

    [19] F. Korouki and A. Razavi, Bounds for the first eigenvalue of $(-Δ-R)$ under the Ricci flow on Bianchi classes, Bull.

    Braz. Math. Soc. (N.S.) 51 (2020), no. 2, 641--651.

    [20] Y. Li, Eigenvalues and entropies under the harmonic-Ricci flow, Pacific J. Math. 267 (2014), no. 1, 141--184.

    [21] L. Ma, Eigenvalue monotonicity for the Ricci-Hamilton flow, Ann. Global Anal. Geom. 29 (2006), no. 3, 287--292.

    [22] J. Milnor, Curvatures of left invariant metrics on Lie groups, Adv. Math. 21 (1976), no. 3, 293--329.

    [23] G. Perelman, The entropy formula for the Ricci flow and it's geometric applications, Arxiv (2002).

    [24] B. Santoro, Introduction to evolution equations in geometry, IMPA, Estrada Doa Castorina, 110 22460-320, Rio de

    Janeiro RJ (2009).

    [25] Y. Z. Wang, Gradient estimates on the weighted p-Laplace heat equation, J. Differ. Equ. 264 (2018), 506--524.

    [26] J. Wu, E. Wang and Y. Zheng, First eigenvalue of p-Laplace operator along the Ricci flow, Ann. Glob. Anal. Geom.

    38 (2010), 27--55.

    [27] N. Zographopoulos, On the principal eigenvalue of degenerate quasilinear elliptic systems, Math. Nachr. 281 (2008),

    1. 9, 1351--1365.