The first Steklov eigenvalue of a free-boundary minimal hypersurface
For $n \ge 3$ let $B^n$ be the open unit ball in $\mathbb{R}^n$, and let $\Sigma^{n-1} \subset \overline{B^n}$ be a smooth compact connected properly embedded minimal hypersurface with nonempty boundary in $S^{n-1}$, meeting $S^{n-1}$ orthogonally along it. With the metric induced from $\mathbb{R}^n$, let $$\sigma_1(\Sigma) = \inf\left\{ \frac{\int_\Sigma |\nabla u|^2}{\int_{\partial\Sigma} u^2} \,:\, u \in H^1(\Sigma),\ \int_{\partial\Sigma} u = 0,\ \int_{\partial\Sigma} u^2 > 0 \right\}$$ be its first positive Steklov eigenvalue. The coordinate functions $x_j$ are harmonic on $\Sigma$ and satisfy $\partial_\eta x_j = x_j$ on $\partial\Sigma$, where $\eta$ is the outward unit conormal, so they are Steklov eigenfunctions with eigenvalue $1$ and $\sigma_1(\Sigma) \le 1$. Fraser and Li proved that $\sigma_1(\Sigma) \ge 1/2$. In $B^3$, $\sigma_1 = 1$ has been proved for free boundary minimal surfaces with suitable symmetries: by McGrath for certain groups generated by reflections in planes, by Kusner and McGrath for the symmetries of many known families, of every genus, and by Seo for genus-zero surfaces with $b$ boundary components and $b$ distinct reflection planes.
Problem. Is $\sigma_1(\Sigma) = 1$ for every $n \ge 3$ and every such $\Sigma$?
Source: Fraser and Li (2014), Conjecture 3.3.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DG — Differential Geometry
- Secondary
- SP
- Keywords
- AIM Open Applied Problems, free boundary minimal surfaces, Steklov eigenvalues
References
@article{fraser2014,
author = {Fraser, Ailana and Li, Martin Man-chun},
title = {Compactness of the space of embedded minimal surfaces with free boundary in three-manifolds with nonnegative {R}icci curvature and convex boundary},
journal = {J. Differential Geom.},
volume = {96},
number = {2},
year = {2014},
pages = {183--200},
doi = {10.4310/jdg/1393424916},
eprint = {1204.6127},
archivePrefix = {arXiv},
}
@article{mcgrath2018,
author = {McGrath, Peter},
title = {A characterization of the critical catenoid},
journal = {Indiana Univ. Math. J.},
volume = {67},
number = {2},
year = {2018},
pages = {889--897},
doi = {10.1512/iumj.2018.67.7251},
eprint = {1603.04114},
archivePrefix = {arXiv},
}
@article{kusner2024,
author = {Kusner, Robert and McGrath, Peter},
title = {On {S}teklov eigenspaces for free boundary minimal surfaces in the unit ball},
journal = {Amer. J. Math.},
volume = {146},
number = {5},
year = {2024},
pages = {1275--1293},
doi = {10.1353/ajm.2024.a937942},
eprint = {2011.06884},
archivePrefix = {arXiv},
}
@article{seo2026,
author = {Seo, Dong-Hwi},
title = {A symmetry condition for genus zero free boundary minimal surfaces attaining the first eigenvalue of one},
journal = {J. Geom. Anal.},
volume = {36},
number = {11},
year = {2026},
pages = {354},
doi = {10.1007/s12220-026-02607-6},
eprint = {2409.14718},
archivePrefix = {arXiv},
}
@misc{aim-free-boundary-first-steklov,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The first Steklov eigenvalue of a free-boundary minimal hypersurface}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/109-free-boundary-first-steklov.md},
note = {Entry 109 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00346, "The first Steklov eigenvalue of a free-boundary minimal hypersurface", clopen.solutions.