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

Abstract 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 $$\mathcal {C}^r$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mi>r</mml:mi> </mml:msup> </mml:math> –Riemannian metrics on M or the set of all $$\mathcal {C}^r$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mi>r</mml:mi> </mml:msup> </mml:math> –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.