# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles 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 deformationsHTML articles powered by AMS MathViewer

by Bruno Colbois and Ahmad El Soufi
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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• 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