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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Eigenvalues of the Laplacian acting on $p$-forms and metric conformal deformations
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by Bruno Colbois and Ahmad El Soufi PDF
Proc. Amer. Math. Soc. 134 (2006), 715-721 Request permission

Abstract:

Let $(M,g)$ be a compact connected orientable Riemannian manifold of dimension $n\ge 4$ and let $\lambda _{k,p} (g)$ be the $k$-th positive eigenvalue of the Laplacian $\Delta _{g,p}=dd^*+d^*d$ acting on differential forms of degree $p$ on $M$. We prove that the metric $g$ can be conformally deformed to a metric $g’$, having the same volume as $g$, with arbitrarily large $\lambda _{1,p} (g’)$ for all $p\in [2,n-2]$. Note that for the other values of $p$, that is $p=0, 1, n-1$ and $n$, one can deduce from the literature that, $\forall k >0$, the $k$-th eigenvalue $\lambda _{k,p}$ is uniformly bounded on any conformal class of metrics of fixed volume on $M$. For $p=1$, we show that, for any positive integer $N$, there exists a metric $g_{_N}$ conformal to $g$ such that, $\forall k\le N$, $\lambda _{k,1} (g_{_N}) =\lambda _{k,0} (g_{_N})$, that is, the first $N$ eigenforms of $\Delta _{g_{_{N},1}}$ are all exact forms.
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Additional Information
  • Bruno Colbois
  • Affiliation: Laboratoire de Mathématiques, Université de Neuchâtel, 13 rue E. Argand, 2007 Neuchâtel, Switzerland
  • MR Author ID: 50460
  • Email: Bruno.Colbois@unine.ch
  • Ahmad El Soufi
  • Affiliation: Laboratoire de Mathématiques et Physique Théorique, Université de Tours, UMR-CNRS 6083, Parc de Grandmont, 37200 Tours, France
  • Email: elsoufi@univ-tours.fr
  • Received by editor(s): July 14, 2004
  • Received by editor(s) in revised form: October 2, 2004
  • Published electronically: July 18, 2005
  • Communicated by: Jozef Dodziuk
  • © Copyright 2005 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 134 (2006), 715-721
  • MSC (2000): Primary 35P15, 58J50, 53C20
  • DOI: https://doi.org/10.1090/S0002-9939-05-08005-6
  • MathSciNet review: 2180889