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Given a smooth compact hypersurface $M$ with boundary $\Sigma=\partial M$, weprove the existence of a sequence $M_j$ of hypersurfaces with the same boundaryas $M$, such that each Steklov eigenvalue $\sigma_k(M_j)$ tends to zero as $j$tends to infinity. The hypersurfaces $M_j$ are obtained from $M$ by a localperturbation near a point of its boundary. Their volumes and diameters arearbitrarily close to those of $M$, while the principal curvatures of theboundary remain unchanged.

Hypersurfaces of Euclidean space with prescribed boundary and small Steklov eigenvalues