Monotonicity formulas for the first eigenvalue of the weighted p-Laplacian under the Ricci-harmonic flow

Let $\Delta _{p,\phi }$ be the weighted p-Laplacian defined on a smooth metric measure space. We study the evolution and monotonicity formulas for the first eigenvalue, $\lambda _{1}=\lambda (\Delta _{p,\phi })$ , of $\Delta _{p,\phi }$ under the Ricci-harmonic flow. We derive some monotonic quantities involving the first eigenvalue, and as a consequence, this shows that $\lambda _{1}$ is monotonically nondecreasing and almost everywhere differentiable along the flow existence.