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

2023/06/10 by Cunha, Cleiton Lira, Gomes, José Nazareno Vieira, Marrocos, Marcus Antônio Mendonça
#47A05 #47A75 (Primary) 47A55 (Secondary) #Differential Geometry (math.DG) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2306.06504

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 C^r--Riemannian metrics on M or the set of all 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

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