On the spectrum of deformations of compact double-sided flat   hypersurfaces
  We study the asymptotic behaviour of the eigenvalues of the Laplace-Beltrami operator on a compact hypersurface in \mathds{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 \varepsilon^2\log\varepsilon and \varepsilon^2. 
Denis Borisov and Pedro Freitas
math.AP math-ph math.MP math.SP