Steklov-isospectral Euclidean domains

The Steklov eigenvalues of a domain measure how it responds to boundary data, much as Laplace eigenvalues measure how a drum vibrates. For drums, Gordon, Webb and Wolpert found differently shaped planar domains with the same spectrum. For the Steklov problem a planar pair was first claimed in August 2026 and awaits confirmation; in three and more dimensions no Euclidean example is known.

Problem

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain with sufficiently regular boundary. Its Steklov eigenvalues $0 = \sigma_0 < \sigma_1 \le \sigma_2 \le \cdots \to \infty$ are the numbers $\sigma$ for which there is a nonzero function $u$ with

$$\Delta u = 0 \ \text{ in } \Omega, \qquad \frac{\partial u}{\partial \nu} = \sigma u \ \text{ on } \partial\Omega,$$

where $\nu$ is the outward unit normal. Equivalently, they are the eigenvalues of the Dirichlet-to-Neumann operator on $\partial\Omega$.

Question. In some dimension $n$, are there two Euclidean domains that are not isometric but have the same Steklov spectrum (with multiplicities)? Or is this impossible?

If the planar construction described below is confirmed, the question stays open for $n \ge 3$, where conformal maps are no longer available. The source also asks the same question for surfaces with boundary; Gordon, Herbrich and Webb have answered that one.

Background

The problem was posed by Leonid Parnovski at the AIM workshop on Steklov eigenproblems. It is the Steklov analogue of Kac's question "Can one hear the shape of a drum?", and Girouard and Polterovich list it among the main open problems in the spectral geometry of the Steklov problem. The Steklov spectrum is determined by the Dirichlet-to-Neumann map, which also appears in inverse problems such as electrical impedance tomography, so spectral rigidity results have independent interest.

What is known

• For surfaces, the number of boundary components and their lengths are determined by the Steklov spectrum (Girouard, Parnovski, Polterovich and Sher, 2014).
• Gordon, Herbrich and Webb (2021) adapted Sunada's method to the Steklov problem, obtaining Steklov-isospectral flat surfaces with boundary and Steklov-isospectral metrics on balls of every dimension above 5. Their planar examples give mixed Steklov–Neumann problems, not the pure Steklov problem on a Euclidean domain.
• For generic curvilinear polygons with angles less than $\pi$, the asymptotics of the Steklov eigenvalues determine the number of vertices, the ordered side lengths and the angles up to an equivalence; for non-generic polygons this fails, and there are non-isometric triangles whose Steklov spectra are asymptotically close (Krymski, Levitin, Parnovski, Polterovich and Sher, 2021).
• Dryden, Gordon, Moreno, Rowlett and Villegas-Blas (2024) showed that for almost every convex polygon there are at most finitely many non-congruent domains with the same Steklov spectrum.
• Hu, Shi and Tang (August 2026, unrefereed preprint) construct pairs of non-congruent, simply connected, strictly convex plane domains with real-analytic boundaries and identical Steklov spectra, arbitrarily close to a disk, using Sunada's method with the Gassmann pair of $GL(3, \mathbb{F}_2)$ and polynomial covers of the disk. If correct, this answers the question in dimension 2.

What would count as progress

• An independent check of the Hu–Shi–Tang construction, for example by computing many Steklov eigenvalues of an explicit pair to high accuracy.
• A pair of non-congruent Euclidean domains in some dimension $n \ge 3$ with identical Steklov spectra, or a proof that none exists.
• A proof that the Steklov spectrum determines a Euclidean domain within a natural class, such as convex polygons or domains with analytic boundary.
• New spectral invariants, from heat-trace or eigenvalue asymptotics, that separate previously indistinguishable domains.

References

1. A. Girouard and I. Polterovich, Spectral geometry of the Steklov problem, survey, arXiv:1411.6567; in Shape Optimization and Spectral Theory, De Gruyter (2017), 120–148.
2. A. Girouard, L. Parnovski, I. Polterovich and D. A. Sher, The Steklov spectrum of surfaces: asymptotics and invariants, Math. Proc. Cambridge Philos. Soc. 157 (2014), 379–389.
3. S. Krymski, M. Levitin, L. Parnovski, I. Polterovich and D. A. Sher, Inverse Steklov spectral problem for curvilinear polygons, Int. Math. Res. Not. IMRN 2021 (2021), 1–37.
4. C. Gordon, P. Herbrich and D. Webb, Steklov and Robin isospectral manifolds, J. Spectral Theory 11 (2021), 39–61; arXiv:1808.10741.
5. E. B. Dryden, C. Gordon, J. Moreno, J. Rowlett and C. Villegas-Blas, The Steklov spectrum of convex polygonal domains I: spectral finiteness, arXiv:2408.01529 (2024).
6. T. Hu, J. Shi and Q. Tang, Strictly convex Steklov-isospectral plane domains, arXiv:2608.10557 (2026).

Source

Imported from AIM Problem Lists stekloveigen 5.2, curated by American Institute of Mathematics. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.