The First Eigenvalue of p-Laplace Operator Under Powers of the mth Mean Curvature Flow

In this paper, we derive the evolution equation for the eigenvalues of p-Laplace operator. Moreover, we show the following main results. Let (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${M^{n}, g(t)), t\in [0,T),}$$\end{document} be a solution of the unnormalized powers of the mth mean curvature flow on a closed manifold and λ1,p(t) be the first eigenvalue of the p-Laplace operator (p ≥ n). At the initial time t = 0, if H > 0, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{ij}\geq \varepsilon Hg_{ij}\quad \left(\frac{1}{p}\leq\epsilon\leq \frac{1}{n}\right),$$\end{document}then λ1,p(t) is nondecreasing and differentiable almost everywhere along the unnormalized powers of the mth mean curvature flow on [0,T). At last, we discuss some interesting monotonic quantities under unnormalized powers of the mth mean curvature flow.