Arbitrarily large Steklov frequency with a prescribed embedded boundary
Let $d \ge 1$ and $m \ge d + 2$. Let $\Sigma \subset \mathbb{R}^m$ be a smooth compact connected embedded $d$-dimensional submanifold without boundary that bounds at least one smooth compact connected embedded $(d+1)$-dimensional submanifold of $\mathbb{R}^m$. Let $\mathcal{F}(\Sigma)$ be the set of all such submanifolds $\Omega \subset \mathbb{R}^m$ with $\partial\Omega = \Sigma$, each with the metric induced from $\mathbb{R}^m$, and let $\sigma_1(\Omega)$ be the first positive Steklov eigenvalue of $\Omega$: the least $\sigma > 0$ for which $\Delta_\Omega u = 0$ in $\Omega$, $\partial_\nu u = \sigma u$ on $\Sigma$ has a nonzero solution. No bound is imposed on the volume or the topology of $\Omega$. Colbois, Girouard and Gittins proved that $\sigma_k(\Omega) \le A_\Sigma |\Omega|\, k^{2/d}$ for every $k \ge 1$ and every $\Omega \in \mathcal{F}(\Sigma)$, with $A_\Sigma$ depending only on $\Sigma$, so fillings with large $\sigma_1$ must have large volume. Colbois, Girouard and Métras showed that a smooth compact hypersurface of Euclidean space can be replaced by hypersurfaces with the same boundary on which every $\sigma_k$ is arbitrarily small. For an abstract compact Riemannian manifold of dimension $d + 1 \ge 3$, Colbois, El Soufi and Girouard made $\sigma_1$ arbitrarily large by conformal changes of the metric that leave it unchanged on the boundary.
Problem. Is $$\sup_{\Omega \in \mathcal{F}(\Sigma)} \sigma_1(\Omega) = \infty$$ for every $d \ge 1$, every $m \ge d + 2$ and every such $\Sigma$?
Source: Colbois, Girouard and Gittins (2019), Remark 1.7; Colbois, Girouard, Gordon and Sher (2024), Open Question 4.52. Stated in this form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- DG
- Keywords
- AIM Open Applied Problems, prescribed boundary, Steklov eigenvalues, submanifolds
References
@article{colbois2019,
author = {Colbois, Bruno and Girouard, Alexandre and Gittins, Katie},
title = {Steklov eigenvalues of submanifolds with prescribed boundary in {E}uclidean space},
journal = {J. Geom. Anal.},
volume = {29},
number = {2},
year = {2019},
pages = {1811--1834},
doi = {10.1007/s12220-018-0063-x},
eprint = {1711.06458},
archivePrefix = {arXiv},
}
@article{colbois2024,
author = {Colbois, Bruno and Girouard, Alexandre and Gordon, Carolyn and Sher, David},
title = {Some recent developments on the {S}teklov eigenvalue problem},
journal = {Rev. Mat. Complut.},
volume = {37},
number = {1},
year = {2024},
pages = {1--161},
doi = {10.1007/s13163-023-00480-3},
eprint = {2212.12528},
archivePrefix = {arXiv},
}
@article{colbois2020,
author = {Colbois, Bruno and Girouard, Alexandre and M{\'e}tras, Antoine},
title = {Hypersurfaces with prescribed boundary and small {S}teklov eigenvalues},
journal = {Canad. Math. Bull.},
volume = {63},
number = {1},
year = {2020},
pages = {46--57},
doi = {10.4153/S000843951900050X},
eprint = {1811.11463},
archivePrefix = {arXiv},
}
@article{colboiselsoufi2019,
author = {Colbois, Bruno and El Soufi, Ahmad and Girouard, Alexandre},
title = {Compact manifolds with fixed boundary and large {S}teklov eigenvalues},
journal = {Proc. Amer. Math. Soc.},
volume = {147},
number = {9},
year = {2019},
pages = {3813--3827},
doi = {10.1090/proc/14426},
}
@misc{aim-fixed-embedded-boundary-steklov,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Arbitrarily large Steklov frequency with a prescribed embedded boundary}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/112-fixed-embedded-boundary-steklov.md},
note = {Entry 112 at commit a153f46, 2026-10-13},
}
Resolution claims
For every smooth compact connected embedded boundary of positive dimension, the first positive Steklov eigenvalue of its smooth embedded fillings is unbounded, provided the ambient codimension is at least two and a filling exists. The boundary remains fixed pointwise, and the fillings carry their induced Euclidean metrics; volume and topology are unrestricted. We give a detailed proof using intrinsic large-eigenvalue …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Open → Solution claimed | The registry's curators | Large Steklov eigenvalues with a prescribed embedded boundary v1 recorded from AIM @ 24edaad, paper/manuscript.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00343, "Arbitrarily large Steklov frequency with a prescribed embedded boundary", clopen.solutions.