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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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