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Eigenvalues of the Laplacian on a compact manifold with density

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.
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https://hal.archives-ouvertes.fr/hal-00870126
Contributor : Ahmad El Soufi Connect in order to contact the contributor
Submitted on : Saturday, October 5, 2013 - 11:53:24 AM
Last modification on : Tuesday, January 11, 2022 - 5:56:08 PM
Long-term archiving on: : Monday, January 6, 2014 - 2:50:19 AM

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  • HAL Id : hal-00870126, version 1
  • ARXIV : 1310.1490

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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⟩

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