Proceedings of the American Mathematical Society

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



Eigenvalues of the Laplacian acting on $p$-forms and metric conformal deformations

Authors: Bruno Colbois and Ahmad El Soufi
Journal: Proc. Amer. Math. Soc. 134 (2006), 715-721
MSC (2000): Primary 35P15, 58J50, 53C20
Published electronically: July 18, 2005
MathSciNet review: 2180889
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Abstract: Let $(M,g)$ be a compact connected orientable Riemannian manifold of dimension $n\ge4$ 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

Ahmad El Soufi
Affiliation: Laboratoire de Mathématiques et Physique Théorique, Université de Tours, UMR-CNRS 6083, Parc de Grandmont, 37200 Tours, France

Keywords: Laplacian, $p$-forms, eigenvalue, conformal deformations.
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
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.