Communications in Analysis and Geometry

Volume 23 (2015)

Number 3

Eigenvalues of the Laplacian on a compact manifold with density

Pages: 639 – 670

DOI: http://dx.doi.org/10.4310/CAG.2015.v23.n3.a6

Authors

Bruno Colbois (Laboratoire de Mathématiques, Université de Neuchâtel, Neuchâtel, Switzerland)

Ahmad El Soufi (Laboratoire de Mathématiques et Physique Théorique, Université Franois Rabelais, Tours, France)

Alessandro Savo (Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Sezione di Matematica, Sapienza Università di Roma, Italy)

Abstract

In this paper, we study the spectrum of the weighted Laplacian (also called Bakry-Émery 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 Schrödinger 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.

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Published 30 January 2015