Eigenvalues of the Laplacian acting on $p$-forms and metric conformal deformations
| dc.creator | Colbois, Bruno | |
| dc.creator | Soufi, Ahmad El | |
| dc.date | 2004-09-15 | |
| dc.date | 2004-09-18 | |
| dc.date.accessioned | 2026-07-07T06:25:32Z | |
| dc.date.available | 2026-07-07T06:25:32Z | |
| dc.description | Let $(M,g)$ be a compact connected orientable Riemannian manifold of dimension $n\ge4$ and let $λ_{k,p} (g)$ be the $k$-th positive eigenvalue of the Laplacian $Δ_{g,p}=dd^*+d^*d$ 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 $λ_{1,p} (g')$ 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 $λ_{k,p}$ 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, $\forall k\le N$, $λ_{k,1} (g_N) =λ_{k,0} (g_N) $, that is, the first $N$ eigenforms of $Δ_{g_N,1}$ are all exact forms. | |
| dc.description | redaction 2003 | |
| dc.identifier | https://arxiv.org/abs/math/0409242 | |
| dc.identifier | http://arxiv.org/abs/math/0409242 | |
| dc.identifier | Proceedings of the American Mathematical Society 134 (3) (2006) 715-721 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/96853 | |
| dc.subject | Differential Geometry | |
| dc.subject | Spectral Theory | |
| dc.subject | 35P15, 58J50, 53C20 | |
| dc.title | Eigenvalues of the Laplacian acting on $p$-forms and metric conformal deformations | |
| dc.type | text |