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

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 ${\Delta }_{g,p}=d{d}^{\ast }+d^{\ast }d$ acting on differential forms of degree $p$ on $M$. We prove that the metric $g$ can be conformally deformed to a metric $g^{\prime }$, having the same volume as $g$, with arbitrarily large ${\lambda }_{1,p}(g^{\prime })$ 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.

redaction 2003