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Eigenvalue estimate for the rough Laplacian on differential forms

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Abstract

In this article, we study the spectrum of the rough Laplacian acting on differential forms on a compact Riemannian manifold (M, g). We first construct on M metrics of volume 1 whose spectrum is as large as desired. Then, provided that the Ricci curvature of g is bounded below, we relate the spectrum of the rough Laplacian on 1-forms to the spectrum of the Laplacian on functions, and derive some upper bound in agreement with the asymptotic Weyl law.

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Authors and Affiliations

  1. Institut de Mathématiques, Université de Neuchâtel, rue Emile Argand 11, 2007, Neuchâtel, Switzerland

    Bruno Colbois

  2. Institut de Mathématiques de Jussieu, UMR CNRS 7586, Université Paris 6, 175 rue du Chevaleret, 75013, Paris, France

    Daniel Maerten

Authors
  1. Bruno Colbois
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  2. Daniel Maerten
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Correspondence to Daniel Maerten.

Additional information

The second author benefits from the ANR grant Géométrie des variétés d’Einstein non compactes ou singulières, n° ANR-06-BLAN-0154-02

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Colbois, B., Maerten, D. Eigenvalue estimate for the rough Laplacian on differential forms. manuscripta math. 132, 399–413 (2010). https://doi.org/10.1007/s00229-010-0352-6

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  • DOI: https://doi.org/10.1007/s00229-010-0352-6

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