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

Let Δp,ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Delta _{p,\phi }$\end{document} 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)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda _{1}=\lambda (\Delta _{p,\phi })$\end{document}, of Δp,ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Delta _{p,\phi }$\end{document} under the Ricci-harmonic flow. We derive some monotonic quantities involving the first eigenvalue, and as a consequence, this shows that λ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda _{1}$\end{document} is monotonically nondecreasing and almost everywhere differentiable along the flow existence.