arXiv research

Sharp upper bound found for Steklov spectrum on revolution submanifolds.

Study finds finitely many non-congruent polygonal domains with same Steklov spectrum.

The paper solves the Steklov spectral inverse problem for conformal metrics.

We prove a lower bound for the $k$-th Steklov eigenvalues in terms of an isoperimetric constant called the $k$-th Cheeger-Steklov constant in three different situations: finite spaces, measurable spaces, and Riemannian manifolds. These lower bounds can be considered as higher order Cheeger type inequalities for the Ste…

Study proves spectral determination of triangles and quadrilaterals, with restrictions on higher-order polygons.

We study the heat trace asymptotics associated with the Steklov eigenvalue problem on a Riemannian manifold with boundary. In particular, we describe the structure of the Steklov heat invariants and compute the first few of them explicitly in terms of the scalar and mean curvatures. This is done by applying the Seeley …

We obtain precise asymptotics for the Steklov eigenvalues on a compact Riemannian surface with boundary. It is shown that the number of connected components of the boundary, as well as their lengths, are invariants of the Steklov spectrum. The proofs are based on pseudodifferential techniques for the Dirichlet-to-Neuma…

We consider how the geometry and topology of a compact $n$-dimensional Riemannian orbifold with boundary relates to its Steklov spectrum. In two dimensions, motivated by work of A. Girouard, L. Parnovski, I. Polterovich and D. Sher in the manifold setting, we compute the precise asymptotics of the Steklov spectrum in t…

Estimates eigenvalues and spectrum for graph substructures using isocapacitary constants.

Study magnetic potentials on Anosov manifolds using spectral data.

Recent developments link Steklov eigenvalues to manifold geometry.

Sharp Steklov eigenvalue estimates for differential forms on manifolds.

Study on biharmonic Steklov problem on differential forms.

This paper is devoted to an inverse Steklov problem for a particular class of n-dimensional manifolds having the topology of a hollow sphere and equipped with a warped product metric. We prove that the knowledge of the Steklov spectrum determines uniquely the associated warping function up to a natural invariance.

Study Dirichlet-to-Neumann maps on manifolds, focusing on covering and total spaces.

New metrics on 3D manifolds with large Steklov eigenvalues.

Study magnetic Steklov eigenvalues on manifolds with boundary.

Study on biharmonic Steklov problems with Neumann boundary conditions and eigenvalue estimates.

Study Cheeger inequalities for Riemannian manifolds with boundary.

Sharp upper bounds found for Steklov eigenvalues of warped products.

Upper bounds for Steklov eigenvalues on manifolds with boundary.

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 Steklo…

The classical inverse problem of recovering a simply connected smooth planar domain from the Steklov spectrum \cite{E} is equivalent to the problem of recovering, up to a conformal equivalence, a positive function $a\in C^\infty({\mathbb S})$ on the unit circle ${\mathbb S}=\{e^{iθ}\}$ from the eigenvalue spectrum of t…

Upper bounds for Steklov eigenvalues in subgraphs of polynomial growth Cayley graphs.

New biharmonic Steklov problem on forms yields eigenvalue estimates.

Higher surgeries preserve Steklov spectra in 3D and above.

The Steklov problem is an eigenvalue problem with the spectral parameter in the boundary conditions, which has various applications. Its spectrum coincides with that of the Dirichlet-to-Neumann operator. Over the past years, there has been a growing interest in the Steklov problem from the viewpoint of spectral geometr…

On any compact manifold of dimension $n\geq3$ with boundary, we prescibe any finite part of the Steklov spectrum whithin a given conformal class. In particular, we prescribe the multiplicity of the first eigenvalues. On a compact surface with boundary, we show that the multiplicity of the $k$-th eigenvalue is bounded i…

Study eigenvalues of magnetic Steklov problem on Riemannian annuli.

Researchers create surfaces with exceptionally high Steklov eigenvalues.

Eigenvalue bounds for forms on warped manifolds studied.

Study of Steklov eigenvalues on degenerating conformal classes.

We consider the class of compact n-dimensional Riemannian manifolds with cylindrical boundary, Ricci curvature bounded below by a given constant and injectivity radius bounded below by a positive constant, away from the boundary. For a manifold M of this class, we introduce a notion of discretization, leading to a grap…

Let M be a compact Riemannian manifold with boundary. Let b>0 be the number of connected components of its boundary. For manifolds of dimension at least 3, we prove that it is possible to obtain an arbitrarily large (b+1)-th Steklov eigenvalue using a smooth conformal perturbation which is supported in a thin neighbour…

Let $M^n=[0,R)\times \mathbb{S}^{n-1}$ be an $n$-dimensional ($n\geq 2$) smooth Riemannian manifold equipped with the warped product metric $g=dr^2+h^2(r)g_{\mathbb{S}^{n-1}}$ and diffeomorphic to a Euclidean ball. Assume that $M$ has strictly convex boundary. First, for the classical Steklov eigenvalue problem, we obt…

Derives integral formula for differential forms on compact spaces with applications.

In this paper we study spectral properties of Dirichlet-to-Neumann map on differential forms obtained by a slight modification of the definition due to Belishev and Sharafutdinov. The resulting operator $Λ$ is shown to be self-adjoint on the subspace of coclosed forms and to have purely discrete spectrum there.We inves…

Physics-Informed Neural Network (PINN) computes the Morse index of the critical catenoid.

We explicitly classify all $S^1$-invariant free boundary minimal annuli and Möbius bands in $\mathbb{B}^n$. This classification is obtained from an analysis of the spectrum of the Dirichlet-to-Neumann map for $S^1$-invariant metrics on the annulus and Möbius band. First, we determine the supremum of the $k$-th normaliz…

Paper finds how Steklov eigenvalues change on graphs and trees.

The paper compares Steklov and Laplacian eigenvalues on graphs.

Upper bound found for Steklov eigenvalues counting function.

Upper bounds found for eigenvalues of weighted Steklov and (p,q)-Laplacian problems.

Eigenvalues of Steklov eigenproblems change predictably with boundary tweaks.

Paper finds a graph Steklov eigenvalue estimate with rigidity results.

Generic metrics on manifolds yield simple Steklov eigenvalues and Morse boundary functions.

Researchers prove rigidity of first conformal Steklov eigenvalue on specific shapes.

Proves inequality for Steklov eigenvalues in hyperbolic space.