Does the first conformal Steklov eigenvalue exceed 2π off the disk?
For a compact connected surface $\Omega$ with boundary and a conformal class $[g]$ of Riemannian metrics on it, let $$\sigma_1^*(\Omega, [g]) = \sup_{h \in [g]} \sigma_1(\Omega, h)\, L(\partial\Omega, h),$$ where $\sigma_1$ is the first nonzero Steklov eigenvalue and $L$ the length of the boundary. By Weinstock's inequality, $\sigma_1^* = 2\pi$ for the disk. Petrides proved that a maximising metric exists in $[g]$ under a gap condition; for the first eigenvalue an affirmative answer below would give it, and so a maximising metric in every such conformal class. The inequality would be the Steklov analogue of Petrides' inequality $\lambda_1^*(M, [g]) > 8\pi$ for the conformal Laplace eigenvalue of a closed surface $M$ other than the sphere.
Problem. Is $\sigma_1^*(\Omega, [g]) > 2\pi$ for every compact connected surface $\Omega$ with boundary that is not diffeomorphic to a disk, and every conformal class $[g]$ on $\Omega$?
Source: Posed by Colbois, Girouard, Gordon and Sher (2024), Open Question 5.39, as the Steklov analogue of Petrides' inequality for conformal Laplace eigenvalues; restated as Conjecture 0.3 in Petrides (2025).
The statement is adapted from doi.org/10.1007/s13163-023-00480-3, whose text is available under CC BY 4.0.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- DG
- Keywords
- Colbois–Girouard–Gordon–Sher Steklov problems, conformal eigenvalues, Riemannian surfaces, Steklov eigenvalues
References
@article{petrides2019,
author = {Petrides, Romain},
title = {Maximizing {S}teklov eigenvalues on surfaces},
journal = {Journal of Differential Geometry},
volume = {113},
number = {1},
pages = {95--188},
year = {2019}
}
@article{petrides2014,
author = {Petrides, Romain},
title = {Existence and regularity of maximal metrics for the first {L}aplace eigenvalue on surfaces},
journal = {Geometric and Functional Analysis},
volume = {24},
number = {4},
pages = {1336--1376},
year = {2014}
}
@misc{matthiesen2020,
author = {Matthiesen, Henrik and Petrides, Romain},
title = {A remark on the rigidity of the first conformal {S}teklov eigenvalue},
year = {2020},
eprint = {2006.04364},
archivePrefix = {arXiv},
note = {The annulus}
}
@misc{petrides2025,
author = {Petrides, Romain},
title = {Isoperimetric inequalities involving {S}teklov eigenvalues on surfaces},
year = {2025},
eprint = {2508.10721},
archivePrefix = {arXiv},
note = {Theorems 0.3 and 0.4: the annulus and the M{\"o}bius band; Conjecture 0.3: the general case}
}
@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 = {Open Question 5.39}
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01023, "Does the first conformal Steklov eigenvalue exceed 2π off the disk?", clopen.solutions.