On the spectrum of deformations of compact double-sided flat hypersurfaces

Abstract

We study the asymptotic behavior of the eigenvalues of the Laplace–Beltrami operator on a compact hypersurface in ${ℝ}^{n+1}$ as it is flattened into a singular double-sided flat hypersurface. We show that the limit spectral problem corresponds to the Dirichlet and Neumann problems on one side of this flat (Euclidean) limit, and derive an explicit three-term asymptotic expansion for the eigenvalues where the remaining two terms are of orders ${\epsilon }^{2}log\epsilon$ and ${\epsilon }^2$.