The Kusner–McGrath conjecture on first Steklov eigenspaces of free boundary minimal hypersurfaces
Let $\Sigma$ be a properly embedded minimal hypersurface in the unit ball $\mathbb B^n$ with free boundary on $\partial\mathbb B^n$. The coordinate functions $x_1, \dots, x_n$ restricted to $\Sigma$ are Steklov eigenfunctions of $\Sigma$ with eigenvalue $1$. The Fraser–Li conjecture asserts that $1$ is the first nonzero Steklov eigenvalue of every such $\Sigma$. In $\mathbb B^3$, Kusner and McGrath proved for surfaces with suitable symmetries, satisfied by many known families, both that conjecture and that the coordinate functions span the whole first eigenspace.
Problem. Does the eigenspace of the first nonzero Steklov eigenvalue of every such $\Sigma \subset \mathbb B^n$ coincide with the span of the coordinate functions of the embedding?
Source: R. Kusner and P. McGrath, On Steklov eigenspaces for free boundary minimal surfaces in the unit ball, Amer. J. Math. 146 (2024), Conjecture 5. Listed as Conjecture 5.43 in Colbois–Girouard–Gordon–Sher (2024).
The statement is adapted from doi.org/10.1007/s13163-023-00480-3, whose text is available under CC BY 4.0.
Classification
- Primary
- DG — Differential Geometry
- Secondary
- SP
- Keywords
- Colbois–Girouard–Gordon–Sher Steklov problems, Fraser–Li conjecture, free boundary minimal surfaces, Steklov eigenvalues
References
@article{kusner2024,
author = {Kusner, Robert and McGrath, Peter},
title = {On {S}teklov eigenspaces for free boundary minimal surfaces in the unit ball},
journal = {American Journal of Mathematics},
volume = {146},
number = {5},
pages = {1275--1293},
year = {2024},
doi = {10.1353/ajm.2024.a937942},
eprint = {2011.06884},
archivePrefix = {arXiv},
note = {Conjecture 5}
}
@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 = {Journal of Differential Geometry},
volume = {96},
number = {2},
pages = {183--200},
year = {2014},
eprint = {1204.6127},
archivePrefix = {arXiv}
}
@misc{parise2026,
author = {Parise, Davide and Zhu, Jonathan J.},
title = {Uniqueness of free boundary minimal annuli},
year = {2026},
eprint = {2609.17512},
archivePrefix = {arXiv},
note = {Theorem 1.3: annuli in $\mathbb{B}^3$}
}
@misc{chodosh2026,
author = {Chodosh, Otis and Gianocca, Matilde},
title = {Free boundary minimal surfaces of genus zero},
year = {2026},
eprint = {2609.13054},
archivePrefix = {arXiv},
note = {The Fraser--Li conjecture for genus zero in $\mathbb{B}^3$}
}
@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 = {Revista Matem{\'a}tica Complutense},
volume = {37},
number = {1},
pages = {1--161},
year = {2024},
doi = {10.1007/s13163-023-00480-3},
eprint = {2212.12528},
archivePrefix = {arXiv},
note = {Conjecture 5.43}
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01024, "The Kusner–McGrath conjecture on first Steklov eigenspaces of free boundary minimal hypersurfaces", clopen.solutions.