The spectrum of the Laplacian in a domain bounded by a flexible polyhedron in $\mathbb R^d$ does not always remain unaltered during the flex

TL;DR

This paper investigates how the eigenvalues of the Laplace operator in domains bounded by flexible polyhedra can change during a flex, challenging the assumption that these spectra are invariant under such deformations.

Contribution

It demonstrates that the Dirichlet and Neumann spectra of the Laplace operator are not necessarily preserved during the flex of a boundary formed by a flexible polyhedron.

Findings

Eigenvalues can vary during the flex of the boundary.

Spectral invariance does not hold for all flexible polyhedral domains.

The change in shape affects the Laplace spectrum during deformation.

Abstract

Being motivated by the theory of flexible polyhedra, we study the Dirichlet and Neumann eigenvalues for the Laplace operator in special bounded domains of Euclidean $d$-space. The boundary of such a domain is an embedded simplicial complex which allows a continuous deformation (a flex), under which each simplex of the complex moves as a solid body and the change in the spatial shape of the domain is achieved through a change of the dihedral angles only. The main result of this article is that both the Dirichlet and Neumann spectra of the Laplace operator in such a domain do not necessarily remain unaltered during the flex of its boundary.