Hypersurfaces of Euclidean space with prescribed boundary and small Steklov eigenvalues

Bruno Colbois; Alexandre Girouard; Antoine Métras

paper

Hypersurfaces of Euclidean space with prescribed boundary and small Steklov eigenvalues

arXiv:1811.11463

Abstract

Given a smooth compact hypersurface $M$ with boundary $Σ = \partial M$ , we prove the existence of a sequence $M_{j}$ of hypersurfaces with the same boundary as $M$ , such that each Steklov eigenvalue $σ_{k} (M_{j})$ tends to zero as $j$ tends to infinity. The hypersurfaces $M_{j}$ are obtained from $M$ by a local perturbation near a point of its boundary. Their volumes and diameters are arbitrarily close to those of $M$ , while the principal curvatures of the boundary remain unchanged.

11 pages, 2 figures