Gradient estimate of an eigenfunction on a compact Riemannian manifold without boundary

Let eλ(x) be an eigenfunction with respect to the Laplace-Beltrami operator ΔM on a compact Riemannian manifold M without boundary: ΔMeλ = λ2eλ. We show the following gradient estimate of eλ: for every λ ≥ 1, there holds \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\lambda\|e_\lambda\|_\infty/C\leq \|\nabla e_\lambda\|_\infty\leq C\lambda\|e_\lambda\|_\infty}$$\end{document}, where C is a positive constant depending only on M.