OP-01009 Open

Existence of a metric maximising a higher normalised Steklov eigenvalue

For a compact surface $\Omega$ with boundary and $k \ge 1$, let $$\sigma_k^*(\Omega) = \sup_g \sigma_k(\Omega, g)\, L(\partial\Omega, g),$$ the supremum running over all Riemannian metrics $g$ on $\Omega$, where $\sigma_k$ is the $k$-th Steklov eigenvalue and $L(\partial\Omega, g)$ the length of the boundary. For $k = 1$, maximising metrics are known to exist on infinitely many surfaces. For the disk and every $k \ge 2$, Fraser and Schoen proved that $\sigma_k^*$ is attained by no smooth metric (Girouard and Polterovich had proved it for $k = 2$). Petrides proved that, on an orientable surface, a smooth maximising metric exists whenever a gap condition $\operatorname{Gap}_k(\Omega) > 0$ holds; for $k \ge 2$ it fails for the disk, and whether it holds on any other surface is open.

Problem. Is there a compact surface $\Omega$ with boundary and an integer $k > 1$ for which a $\sigma_k$-maximising metric exists, that is, for which $\sigma_k^*(\Omega)$ is attained?

Source: Posed by Colbois, Girouard, Gordon and Sher (2024), Open Question 5.31, after Fraser and Schoen (2020) and Petrides (2019).

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, eigenvalue optimisation, Riemannian surfaces, Steklov eigenvalues

References

@article{fraser2020,
  author = {Fraser, Ailana and Schoen, Richard},
  title = {Some results on higher eigenvalue optimization},
  journal = {Calculus of Variations and Partial Differential Equations},
  volume = {59},
  number = {5},
  pages = {Paper No. 151},
  year = {2020}
}

@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{girouard2010,
  author = {Girouard, Alexandre and Polterovich, Iosif},
  title = {On the {H}ersch--{P}ayne--{S}chiffer inequalities for {S}teklov eigenvalues},
  journal = {Functional Analysis and Its Applications},
  volume = {44},
  number = {2},
  pages = {106--117},
  year = {2010}
}

@misc{petrides2025,
  author = {Petrides, Romain},
  title = {Isoperimetric inequalities involving {S}teklov eigenvalues on surfaces},
  year = {2025},
  eprint = {2508.10721},
  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 = {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 5.31}
}

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-01009, "Existence of a metric maximising a higher normalised Steklov eigenvalue", clopen.solutions.