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The Steklov and Laplacian spectra of Riemannian manifolds with boundary
Given two compact Riemannian manifolds $M_1$ and $M_2$ such that their respective boundaries $\Sigma_1$ and $\Sigma_2$ admit neighbourhoods $\Omega_1$ and $\Omega_2$ which are isometric, we prove the existence of a constant $C$ such that $
Steklov Eigenvalues of Submanifolds with Prescribed Boundary in Euclidean Space
We obtain upper and lower bounds for Steklov eigenvalues of submanifolds with prescribed boundary in Euclidean space. A very general upper bound is proved, which depends only on the geometry of the fixed boundary and on the measure of the interior. Sharp lower bounds are given for hypersurfaces of revolution with connected boundary: we prove that each eigenvalue is uniquely minimized by the ball. We also observe that each surface of revolution with connected boundary is Steklov isospectral to the disk.
Uniform stability of the Dirichlet spectrum for rough perturbations
Hypersurfaces with prescribed boundary and small Steklov eigenvalues
iven a smooth compact hypersurface $M$ with boundary $\Sigma=\partial M$, we prove the existence of a sequence $M_j$ of hypersurfaces with the same boundary as $M$, 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 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.
The Steklov spectrum and coarse discretizations of manifolds with boundary
Given $\kappa, r_0>0$ and $n\in\N$, we consider the class $\mathcal{M}=\mathcal{M}(\kappa,r_0,n)$ of compact $n$-dimensional Riemannian manifolds with cylindrical boundary, Ricci curvature bounded below by $-(n-1)\kappa$ and injectivity radius bounded below by $r_0$ away from the boundary. For a manifold $M\in\mathcal{M}$ we introduce a notion of discretization, leading to a graph with boundary which is roughly isometric to $M$, with constants depending only on $\kappa,r_0,n$. In this context, we prove a uniform spectral comparison inequality between the Steklov eigenvalues of a manifold $M\in\mathcal{M}$ and those of its discretization. Some applications to the construction of sequences of surfaces with boundary of fixed length and with arbitrarily large Steklov spectral gap $\sigma_2-\sigma_1$ are given. In particular, we obtain such a sequence for surfaces with connected boundary. The applications are based on the construction of graph-like surfaces which are obtained from sequences of graphs with good expansion properties.
Isoperimetric control of the spectrum of a compact hypersurface
Compact manifolds with fixed boundary and large Steklov eigenvalues
Let $(M,g)$ be a compact Riemannian manifold with boundary. Let $b>0$ be the number of connected components of its boundary. For manifolds of dimension $\geq 3$, we prove that for $j=b+1$ it is possible to obtain an arbitrarily large Steklov eigenvalue $\sigma_j(M,e^\delta g)$ using a conformal perturbation $\delta\in C^\infty(M)$ which is supported in a thin neighbourhood of the boundary, with $\delta=0$ on the boundary. For $j\leq b$, it is also possible to obtain arbitrarily large eigenvalues, but the conformal factor must spread throughout the interior of $M$. This is in stark contrast with the situation for the eigenvalues of the Laplace operator, for which the supremum is bounded in each fixed conformal class. In fact, when working in a fixed conformal class, it is known that the volume of $(M,e^\delta g)$ has to tend to infinity in order for some $\sigma_j$ to become arbitrarily large. We also prove that it is possible to obtain large eigenvalues while keeping different boundary components arbitrarily close to each others, by constructing a convenient Riemannian submersion.
Isoperimetric control of the Steklov spectrum
We prove that the normalized Steklov eigenvalues of a bounded domain in a complete Riemannian manifold are bounded above in terms of the inverse 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 uniforml bounded above. On a compact surface with boundary, we obtain uniform bounds for the normalized Steklov eigenvalues in terms of the genus. We also establish a relationship between the Steklov eigenvalues of a domain and the eigenvalues of the Laplace-Beltrami operator on its boundary hypersurface.