Abstract
We prescribe, for a Euclidean domain with given topology, any finite part of the spectrum of the Hodge and de Rham Laplacian, with multiplicity 1 or 2. The domains which yield this result do not depend on the degree of the differential forms when it is between 2 and n−1. Moreover we also prescribe the volume in this case. We get a similar result for a compact manifold. The proof relies on a generalization to differential forms of the “Cheeger's dumbbell balls” as well as on a result of convergence of the spectrum for manifolds connected together with cylinders of fixed lengths and radii which tend to 0.
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