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

TL;DR

This paper investigates how the eigenvalues of the Laplace-Beltrami operator behave when a compact hypersurface in Euclidean space is flattened into a double-sided flat shape, revealing the limit spectral problem and providing detailed asymptotic expansions.

Contribution

It introduces a precise asymptotic analysis of eigenvalues during the flattening process, connecting the spectral problem to Dirichlet and Neumann conditions on the limit shape.

Findings

Limit spectral problem corresponds to Dirichlet and Neumann problems on one side.

Explicit three-term asymptotic expansion for eigenvalues.

Remaining terms are of orders b5^2 extlogb5 and b5^2.

Abstract

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.