Mathematics – Differential Geometry
Scientific paper
2004-09-15
Proceedings of the American Mathematical Society 134 (3) (2006) 715-721
Mathematics
Differential Geometry
redaction 2003
Scientific paper
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.
Colbois Bruno
Soufi Ahmad El
No associations
LandOfFree
Eigenvalues of the Laplacian acting on $p$-forms and metric conformal deformations does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Eigenvalues of the Laplacian acting on $p$-forms and metric conformal deformations, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Eigenvalues of the Laplacian acting on $p$-forms and metric conformal deformations will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-621803