Uniqueness of the critical catenoid
Let $B^3$ be the open unit ball in $\mathbb{R}^3$ and $S^2 = \partial B^3$. A free boundary minimal annulus is a smooth compact connected minimal surface $\Sigma \subset \overline{B^3}$, properly embedded and homeomorphic to an annulus, with its interior in $B^3$ and its boundary in $S^2$, meeting $S^2$ orthogonally. The critical catenoid is the free boundary minimal annulus $$X(t,\theta) = a\,(\cosh t \cos\theta,\ \cosh t \sin\theta,\ t), \qquad |t| \le t_0,\quad \theta \in \mathbb{R}/2\pi\mathbb{Z},$$ where $t_0 > 0$ solves $t_0 \tanh t_0 = 1$ and $a = (\cosh^2 t_0 + t_0^2)^{-1/2}$. The coordinate functions of a free boundary minimal surface in the ball are Steklov eigenfunctions with eigenvalue $1$. Fraser and Schoen proved that a free boundary minimal annulus in the ball whose coordinate functions are first Steklov eigenfunctions is the critical catenoid, up to congruence. McGrath proved uniqueness for annuli symmetric under reflection through the three coordinate planes, Kusner and McGrath for annuli invariant under the antipodal map, and Espinar and Marín for annuli whose support function has infinitely many critical points. Embeddedness cannot be dropped: Fernández, Hauswirth and Mira constructed immersed, non-embedded free boundary minimal annuli in $B^3$ other than the critical catenoid.
Problem. Is every free boundary minimal annulus $\Sigma$ in $B^3$ carried onto the critical catenoid by a rotation?
Source: Fraser and Li (2014), Conjecture 1.1.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DG — Differential Geometry
- Secondary
- AP, SP
- Keywords
- AIM Open Applied Problems, critical catenoid, 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{fraserschoen2016,
author = {Fraser, Ailana and Schoen, Richard},
title = {Sharp eigenvalue bounds and minimal surfaces in the ball},
journal = {Invent. Math.},
volume = {203},
number = {3},
year = {2016},
pages = {823--890},
doi = {10.1007/s00222-015-0604-x},
}
@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{espinar2026,
author = {Espinar, Jos{\'e} M. and Mar{\'i}n, Diego A.},
title = {An overdetermined eigenvalue problem and the critical catenoid conjecture},
journal = {J. Eur. Math. Soc. (JEMS)},
year = {2026},
doi = {10.4171/JEMS/1783},
eprint = {2310.06705},
archivePrefix = {arXiv},
}
@article{fernandez2023,
author = {Fern{\'a}ndez, Isabel and Hauswirth, Laurent and Mira, Pablo},
title = {Free boundary minimal annuli immersed in the unit ball},
journal = {Arch. Ration. Mech. Anal.},
volume = {247},
number = {6},
year = {2023},
pages = {108},
doi = {10.1007/s00205-023-01943-z},
eprint = {2208.14998},
archivePrefix = {arXiv},
}
@misc{aim-critical-catenoid-uniqueness,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Uniqueness of the critical catenoid}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/108-critical-catenoid-uniqueness.md},
note = {Entry 108 at commit a153f46, 2026-10-13},
}
Resolution claims
We prove that every smooth properly embedded free-boundary minimal annulus in the unit ball is congruent to the critical catenoid. We also classify positive solutions of $(Δ_{\mathbb{S}^2}+2)u=0$ on smooth spherical annuli with $u=0$ and $|\nabla u|=1$ on the boundary; their one-homogeneous extensions are the axially symmetric Alt-Caffarelli cones. Lastly, we also treat the case of free-boundary minimal annuli in spherical …
We prove that any embedded genus zero free boundary minimal surface in the ball has first Steklov eigenvalue $1$. This implies the uniqueness of the critical catenoid.
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-15 | Needs verification → Solution claimed | — | Recorded here: arXiv:2609.13054v1. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00318, "Uniqueness of the critical catenoid", clopen.solutions.