Hadamard-type variation formulae for the eigenvalues of a class of second-order elliptic operators and its applications

We use variational methods to derive Hadamard-type formulae for the eigenvalues of a class of elliptic operators on a compact Riemannian manifold $M$. We then apply the latter in the following context. Consider a family of elliptic operators which is parametrized by either the set of all $\mathcal{C}^r$--Riemannian metrics on $M$ or the set of all $\mathcal{C}^r$--diffeomorphisms on a domain into $M$. In either case we prove that if a subset of the parametrizations set yields a simple spectrum of the operator, then it is necessarily a generic subset. We also analyse the behavior of the eigenvalues when the metric evolves along the Ricci flow on a closed Riemannian manifold, and we prove, under a suitable hypothesis, that they increase