LAUSR

Let Δp,ϕ$\Delta _{p,\phi }$ be the weighted p-Laplacian defined on a smooth metric measure space. We study the evolution and monotonicity formulas for the first eigenvalue, λ1=λ(Δp,ϕ)$\lambda _{1}=\lambda (\Delta _{p,\phi })$, of Δp,ϕ$\Delta _{p,\phi }$ under the Ricci-harmonic flow. We derive some monotonic quantities involving the first eigenvalue, and as a consequence, this shows that λ1$\lambda _{1}$ is monotonically nondecreasing and almost everywhere differentiable along the flow existence.