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

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.