Monotonicity of First Eigenvalues along the Yamabe Flow

We construct some nondecreasing quantities associated to the first eigenvalue of −Δφ+cR(c≥12(n−2)/(n−1))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ - {{\rm{\Delta }}_\varphi } + cR\left( {c \ge {1 \over 2}\left( {n - 2} \right)/\left( {n - 1} \right)} \right)$$\end{document} along the Yamabe flow, where Δφ is the Witten-Laplacian operator with a C2 function φ. We also prove a monotonic result on the first eigenvalue of −Δφ+14(n/(n−1))R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ - {{\rm{\Delta }}_\varphi } + {1 \over 4}\left( {n/\left( {n - 1} \right)} \right)R$$\end{document} along the Yamabe flow. Moreover, we establish some nondecreasing quantities for the first eigenvalue of −Δφ + cRa with a ∈ (0,1) along the Yamabe flow.