## 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.## References

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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