Some universal inequalities of eigenvalues and upper bounds for the $L_{\mathrm{\infty }}$ norm of eigenfunctions of the Laplacian

In this short survey, we derive some weyl-type universal inequalities of eigenvalues of the Laplacian on a closed Riemannian manifold of nonnegative Ricci curvature. We also give upper bounds for the $L_{\mathrm{\infty }}$ norm of eigenfunctions of the Laplacian in the same setting. A detailed proof of these results did not seem to appear in the literature but the results follow from a simple combination of Milman’s work and Cheng-Li’s work.