The spectral gap of graphs and Steklov eigenvalues on surfaces

Using expander graphs, we construct a sequence of smooth compact surfaces with boundary of perimeter N, and with the first non-zero Steklov eigenvalue uniformly bounded away from zero. This answers a question which was raised in [9]. The genus grows linearly with N, this is the optimal growth rate.Comment: 9 pages, 1 figur

The Steklov and Laplacian spectra of Riemannian manifolds with boundary

Given two compact Riemannian manifolds with boundary $M_1$ and $M_2$ such that their respective boundaries $\Sigma_1$ and $\Sigma_2$ admit neighborhoods $\Omega_1$ and $\Omega_2$ which are isometric, we prove the existence of a constant $C$, which depends only on the geometry of $\Omega_1\cong\Omega_2$, such that $|\sigma_k(M_1)-\sigma_k(M_2)|\leq C$ for each $k\in\mathbb{N}$. This follows from a quantitative relationship between the Steklov eigenvalues $\sigma_k$ of a compact Riemannian manifold $M$ and the eigenvalues $\lambda_k$ of the Laplacian on its boundary. Our main result states that the difference $|\sigma_k-\sqrt{\lambda_k}|$ is bounded above by a constant which depends on the geometry of $M$ only in a neighborhood of its boundary. The proofs are based on a Pohozaev identity and on comparison geometry for principal curvatures of parallel hypersurfaces. In several situations, the constant $C$ is given explicitly in terms of bounds on the geometry of $\Omega_1\cong\Omega_2$.Comment: 31 pages, 1 figur

Isoperimetric control of the spectrum of a compact hypersurface

Upper bounds for the eigenvalues of the Laplace-Beltrami operator on a hypersurface bounding a domain in some ambient Riemannian manifold are given in terms of the isoperimetric ratio of the domain. These results are applied to the extrinsic geometry of isometric embeddings.Comment: To appear in Journal f\"ur die reine und angewandte Mathematik (Crelle's Journal

Isoperimetric control of the Steklov spectrum

Let N be a complete Riemannian manifold of dimension n+1 whose Riemannian metric g is conformally equivalent to a metric with non-negative Ricci curvature. The normalized Steklov eigenvalues of a bounded domain in N are bounded above in terms of the isoperimetric ratio of the domain. Consequently, the normalized Steklov eigenvalues of a bounded domain in Euclidean space, hyperbolic space or a standard hemisphere are uniformly bounded above. On a compact surface with boundary, the normalized Steklov eigenvalues are uniformly bounded above in terms of the genus. We also obtain a relationship between the Steklov eigenvalues of a domain and the eigenvalues of the Laplace-Beltrami operator on its bounding hypersurface