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

Abstract

Let ( M , g ) (M,g) be a compact connected orientable Riemannian manifold of dimension n ≥ 4 n\ge 4 and let λ k , p ( g ) \lambda _{k,p} (g) be the k k -th positive eigenvalue of the Laplacian Δ g , p = d d ∗ + d ∗ d \Delta _{g,p}=dd^*+d^*d acting on differential forms of degree p p on M M . We prove that the metric g g can be conformally deformed to a metric g ′ g’ , having the same volume as g g , with arbitrarily large λ 1 , p ( g ′ ) \lambda _{1,p} (g’) for all p ∈ [ 2 , n − 2 ] p\in [2,n-2] . Note that for the other values of p p , that is p = 0 , 1 , n − 1 p=0, 1, n-1 and n n , one can deduce from the literature that, ∀ k > 0 \forall k >0 , the k k -th eigenvalue λ k , p \lambda _{k,p} is uniformly bounded on any conformal class of metrics of fixed volume on M M . For p = 1 p=1 , we show that, for any positive integer N N , there exists a metric g N g_{_N} conformal to g g such that, ∀ k ≤ N \forall k\le N , λ k , 1 ( g N ) = λ k , 0 ( g N ) \lambda _{k,1} (g_{_N}) =\lambda _{k,0} (g_{_N}) , that is, the first N N eigenforms of Δ g N , 1 \Delta _{g_{_{N},1}} are all exact forms.