Eigenvalues of the Laplacian on a compact manifold with density

TL;DR

This paper investigates the eigenvalues of the weighted Laplacian on compact manifolds, providing upper bounds dependent on the density function and demonstrating sharpness through examples, with applications to related operators.

Contribution

It introduces new upper bounds for the weighted Laplacian's eigenvalues based on density norms and extends classical inequalities to the weighted setting.

Findings

Derived eigenvalue bounds consistent with Weyl's law

Examples showing sharpness of density dependence

Extended inequalities to Schrödinger and Hodge Laplacians

Abstract

In this paper, we study the spectrum of the weighted Laplacian (also called Bakry-Emery or Witten Laplacian) $L_\sigma$ on a compact, connected, smooth Riemannian manifold $(M,g)$ endowed with a measure $\sigma dv_g$. First, we obtain upper bounds for the $k-$th eigenvalue of $L_{\sigma}$ which are consistent with the power of $k$ in Weyl's formula. These bounds depend on integral norms of the density $\sigma$, and in the second part of the article, we give examples showing that this dependence is, in some sense, sharp. As a corollary, we get bounds for the eigenvalues of Laplace type operators, such as the Schr\"{o}dinger operator or the Hodge Laplacian on $p-$forms. In the special case of the weighted Laplacian on the sphere, we get a sharp inequality for the first nonzero eigenvalue which extends Hersch's inequality.