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

TL;DR

This survey presents universal inequalities for Laplacian eigenvalues and bounds on eigenfunction norms on manifolds with nonnegative Ricci curvature, combining existing results for new insights.

Contribution

It provides new universal inequalities and upper bounds for eigenvalues and eigenfunctions of the Laplacian on specific Riemannian manifolds, with detailed proofs.

Findings

Universal inequalities for Laplacian eigenvalues

Upper bounds for eigenfunction $L_{}$ norms

Results derived from Milman and Cheng-Li's work

Abstract

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 $L_{\infty}$ 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.