Eigenvalues of the Laplacian on a compact manifold with density

Bruno Colbois (1) , Ahmad El Soufi (2) , Alessandro Savo (3)

1 Laboratoire de Mathématiques
2 LMPT - Laboratoire de Mathématiques et Physique Théorique
3 MeMoMat - Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate

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.

Mots clés: Upper bound; Eigenvalue; Weighted Laplacian; Witten Laplacian; Manifold with density

Domaines: Géométrie métrique [math.MG]

Bruno Colbois, Ahmad El Soufi, Alessandro Savo. Eigenvalues of the Laplacian on a compact manifold with density. Communications in Analysis and Geometry, 2015, 23 (3), pp.639--670. ⟨hal-00870126⟩