Spectra of Manifolds with Holes and the Dirichlet-to-Neumann Operator of ∆S2 + 2 on a Sphere with Disks Removed

Specialeforsvar ved Martin Ravn Christiansen

Titel: Spectra of Manifolds with Holes and the Dirichlet-to-Neumann  Operator of ∆S₂ + 2 on a Sphere with Disks Removed

Abstract: This thesis examines the behavior of the eigenvalues of the Laplace operator of a compact Riemannian manifold with holes as the radius of the holes goes to 0. This behavior depends on the boundary conditions one assigns to the boundary of the holes and we study both the case of Dirichlet and Neumann boundary conditions.
The case of Dirichlet boundary conditions has been studied previously in the literature and an asymptotic formula for the perturbed eigenvalues has been given. We derive an explicit bound on the rate of convergence of the perturbed eigenvalues to the unperturbed eigenvalues and rederive the asymptotic formula under the additional assumption that the manifold has constant sectional curvature. We also study the convergence of the asso-ciated eigenfunctions and in particular present a result on the convergence of eigen-functions associated with degenerate eigenvalues.
The case of Neumann boundary conditions is less studied in the literature although again an asymptotic formula has been given. We derive a general upper bound on the perturbed eigenvalues in terms of the unperturbed eigenvalues and for dimensions greater than 4 also derive a lower bound which in particular implies convergence of the eigenvalues. We also reproduce the asymptotic formula in a special case. Finally we present an application of these results. We consider the Dirichlet-to-Neumann operator ∆S₂ + 2 on a round sphere with disks removed, which is an operator arising in the study of minimal surfaces. By applying theeigenvalue and eigenvector convergence results we characterize the leading asymptotics of this operator as the radius of the disks goes to 0.

Vejleder: Niels Martin Møller
Censor:   Poul Hjorth, DTU