#### Vol. 6, No. 5, 2013

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

### Denis Borisov and Pedro Freitas

Vol. 6 (2013), No. 5, 1051–1088
##### 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 ${\epsilon }^{2}$.

##### Keywords
Laplace–Beltrami operator, eigenvalue, flat manifolds
Primary: 35P15
Secondary: 35J05