Une inégalité du type Payne-Polya-Weinberger pour le laplacien brut

Let us consider a riemannian vector bundle $E$ with compact basis $(M,g)$ and the rough laplacian $\bar{\Delta}$ associated to a connection $D$ on $E$. We prove that the eigenvalues of $\bar{\Delta}$ are bounded above by a function of the first eigenvalue and of the geometry of $(M,g)$, but independently of the choice of the connection $D$.