2015/10/23 by Gomes, J. N. V., Marrocos, M. A. M., Mesquita, R. R.
#Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1510.07076
We consider an analytic family of Riemannian metrics on a compact smooth manifold M. We assume the Dirichlet boundary condition for the η-Laplacian and obtain Hadamard type variation formulas for analytic curves of eigenfunctions and eigenvalues. As an application, we show that for a subset of all Cr Riemannian metrics Mr on M, all eigenvalues of the η-Laplacian are generically simple, for 2≤ r< ∞. This implies the existence of a residual set of metrics in Mr, which makes the spectrum of the η-Laplacian simple. Likewise, we show that there exists a residual set of drifting functions η in the space Fr of all Cr functions on M, which makes again the spectrum of the η-Laplacian simple, for 2≤ r< ∞. Besides, we give a precise information about the complementary of these residual sets, as well as about the structure of the set of deformations of a Riemannian metric (respectively of the set of deformations of a drifting function) which preserves double eigenvalues. Moreover, we consider a family of perturbations of a domain in a Riemannian manifold and obtain Hadamard type formulas for the eigenvalues of the η-Laplacian in this case. We also establish generic properties of eigenvalues in this context.