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 $n\ge4$ 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.