Variation of the first eigenvalue of $(p,q)$-Laplacian along the Ricci-harmonic flow

Document Type : Research Paper


Department of Pure Mathematics, Faculty of Sciences, Imam Khomeini International University, Qazvin, Iran.


In this paper, we study monotonicity for the first eigenvalue of a class of $(p,q)$-Laplacian. We find the first variation formula for the first eigenvalue of $(p,q)$-Laplacian on a closed Riemannian manifold  evolving by the Ricci-harmonic flow and construct various monotonic quantities by imposing some conditions on initial manifold.