Steklov-isospectral Euclidean domains

Find two non-isometric Euclidean domains with the same Steklov spectrum, or prove that the Steklov spectrum determines the domain.

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1	# Steklov-isospectral Euclidean domains
2
3	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.
4
5	## Problem
6
7	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
8
9	$$\Delta u = 0 \ \text{ in } \Omega, \qquad \frac{\partial u}{\partial \nu} = \sigma u \ \text{ on } \partial\Omega,$$
10
11	where $\nu$ is the outward unit normal. Equivalently, they are the eigenvalues of the Dirichlet-to-Neumann operator on $\partial\Omega$.
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13	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?
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15	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.
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17	## Background
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19	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.
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21	## What is known
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23	- For surfaces, the number of boundary components and their lengths are determined by the Steklov spectrum (Girouard, Parnovski, Polterovich and Sher, 2014).
24	- 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.
25	- 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).
26	- 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.
27	- 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.
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29	## What would count as progress
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31	- An independent check of the Hu–Shi–Tang construction, for example by computing many Steklov eigenvalues of an explicit pair to high accuracy.
32	- A pair of non-congruent Euclidean domains in some dimension $n \ge 3$ with identical Steklov spectra, or a proof that none exists.
33	- A proof that the Steklov spectrum determines a Euclidean domain within a natural class, such as convex polygons or domains with analytic boundary.
34	- New spectral invariants, from heat-trace or eigenvalue asymptotics, that separate previously indistinguishable domains.
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36	## References
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38	1. A. Girouard and I. Polterovich, [Spectral geometry of the Steklov problem](https://arxiv.org/abs/1411.6567), survey, arXiv:1411.6567; in Shape Optimization and Spectral Theory, De Gruyter (2017), 120–148.
39	2. A. Girouard, L. Parnovski, I. Polterovich and D. A. Sher, [The Steklov spectrum of surfaces: asymptotics and invariants](https://doi.org/10.1017/S030500411400036X), Math. Proc. Cambridge Philos. Soc. 157 (2014), 379–389.
40	3. S. Krymski, M. Levitin, L. Parnovski, I. Polterovich and D. A. Sher, [Inverse Steklov spectral problem for curvilinear polygons](https://doi.org/10.1093/imrn/rnaa200), Int. Math. Res. Not. IMRN 2021 (2021), 1–37.
41	4. C. Gordon, P. Herbrich and D. Webb, [Steklov and Robin isospectral manifolds](https://doi.org/10.4171/jst/335), J. Spectral Theory 11 (2021), 39–61; [arXiv:1808.10741](https://arxiv.org/abs/1808.10741).
42	5. E. B. Dryden, C. Gordon, J. Moreno, J. Rowlett and C. Villegas-Blas, [The Steklov spectrum of convex polygonal domains I: spectral finiteness](https://arxiv.org/abs/2408.01529), arXiv:2408.01529 (2024).
43	6. T. Hu, J. Shi and Q. Tang, [Strictly convex Steklov-isospectral plane domains](https://arxiv.org/abs/2608.10557), arXiv:2608.10557 (2026).
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45	## Source
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47	Imported from [AIM Problem Lists stekloveigen 5.2](http://aimpl.org/stekloveigen/5/), 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.