Unbounded first Steklov multiplicity with fixed boundary count
For a smooth compact connected orientable Riemannian surface $(M, g)$ with nonempty boundary, let $\sigma_1(M, g)$ be the first positive eigenvalue of the Steklov problem $\Delta_g u = 0$ in $M$, $\partial_\nu u = \sigma u$ on $\partial M$, without weights, and let $$m_1(M, g) = \dim\{u : \Delta_g u = 0,\ \partial_\nu u = \sigma_1(M, g)\, u\}$$ be its multiplicity. Karpukhin, Kokarev and Polterovich proved that $m_1(M, g) \le 4\gamma + 3$ when $M$ has genus $\gamma$, so along any sequence with $m_1 \to \infty$ the genus tends to infinity. Audet-Beaumont constructed, for each prime $p$, such a surface of genus $(p-1)^2$ with $2p$ boundary components and $m_1 \ge p - 1$; in his surfaces the number of boundary components grows with the multiplicity.
Problem. Does there exist a fixed integer $b \ge 1$ and a sequence of smooth compact connected orientable Riemannian surfaces $(M_j, g_j)$, each with exactly $b$ boundary components, such that $m_1(M_j, g_j) \to \infty$? The genus and the metric may vary with $j$; $b$ may not.
Source: Audet-Beaumont (2026), Remark 1.3. The question without a fixed number of boundary components, with boundary densities allowed, is Karpukhin, Kokarev and Polterovich's (2014), Question 1.4.2.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- DG
- Keywords
- AIM Open Applied Problems, eigenvalue multiplicities, Riemannian surfaces, Steklov eigenvalues
References
@article{audetbeaumont2026,
author = {Audet-Beaumont, Samuel},
title = {Constructing surfaces with first {S}teklov eigenvalue of arbitrarily large multiplicity},
journal = {Canad. Math. Bull.},
volume = {69},
number = {2},
year = {2026},
pages = {572--586},
doi = {10.4153/S0008439525101288},
eprint = {2412.07692},
archivePrefix = {arXiv},
}
@article{karpukhin2014,
author = {Karpukhin, Mikhail and Kokarev, Gerasim and Polterovich, Iosif},
title = {Multiplicity bounds for {S}teklov eigenvalues on {R}iemannian surfaces},
journal = {Ann. Inst. Fourier (Grenoble)},
volume = {64},
number = {6},
year = {2014},
pages = {2481--2502},
doi = {10.5802/aif.2918},
eprint = {1209.4869},
archivePrefix = {arXiv},
}
@misc{aim-steklov-fixed-boundary-multiplicity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Unbounded first Steklov multiplicity with fixed boundary count}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/111-steklov-fixed-boundary-multiplicity.md},
note = {Entry 111 at commit a153f46, 2026-10-13},
}
Resolution claims
Let $0<a_1\le\cdots\le a_N$ and $H>a_N$. On every compact connected orientable surface with $b\ge1$ boundary components and genus at least an explicit integer $\Gamma(N,b)$, we construct a smooth metric whose first $N$ positive Steklov eigenvalues are $a_1,\ldots,a_N$ and whose next eigenvalue exceeds $H$. Repetitions are allowed. For connected boundary, $\Gamma(N,1)=\lceil N(N-1)/4\rceil$. This is the least genus permitted by the complete-graph band …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-20 | Open → Solution claimed | The registry's curators | Prescribed Steklov spectra and multiplicities on surfaces v1 recorded from Zenodo:10.5281/zenodo.22949635. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00390, "Unbounded first Steklov multiplicity with fixed boundary count", clopen.solutions.