OP-01025 Open

Strict monotonicity of normalised conformal Steklov eigenvalues

Let $\Omega$ be a compact connected manifold of dimension $d + 1$ with boundary, and for a Riemannian metric $g$ on $\Omega$ let $\bar\sigma_k(\Omega, g) = \sigma_k(\Omega, g)\, |\Omega|_g^{(1-d)/(d+1)}\, |\partial\Omega|_g$, where $\sigma_k$ is the $k$-th Steklov eigenvalue ($\sigma_0 = 0$). For a conformal class $[g]$ let $$\bar\sigma_k^*(\Omega, [g]) = \sup_{h \in [g]} \bar\sigma_k(\Omega, h).$$ For the Laplacian on closed manifolds, Colbois and El Soufi proved the analogous strict inequalities between consecutive conformal eigenvalues. An affirmative answer below would imply that $(g, 1)$ is a conformally critical pair, in the sense of Karpukhin and Métras, for every metric $g$ maximising $\bar\sigma_k$ in its conformal class.

Problem. Is $\bar\sigma_k^*(\Omega, [g]) > \bar\sigma_{k-1}^*(\Omega, [g])$ for every compact connected manifold $\Omega$ with boundary, every conformal class $[g]$ on $\Omega$ and every $k \ge 1$?

Source: Posed by Colbois, Girouard, Gordon and Sher (2024), Open Question 5.56, as the Steklov analogue of a theorem of Colbois and El Soufi (2003).

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, Steklov eigenvalues

References

@article{colbois2003,
  author = {Colbois, Bruno and El Soufi, Ahmad},
  title = {Extremal eigenvalues of the {L}aplacian in a conformal class of metrics: the `conformal spectrum'},
  journal = {Annals of Global Analysis and Geometry},
  volume = {24},
  number = {4},
  pages = {337--349},
  year = {2003},
  doi = {10.1023/A:1026257431539}
}

@misc{petrides2025,
  author = {Petrides, Romain},
  title = {Isoperimetric inequalities involving {S}teklov eigenvalues on surfaces},
  year = {2025},
  eprint = {2508.10721},
  archivePrefix = {arXiv},
  note = {Theorem 0.2 (disk gluing in a conformal class), from which the dimension-two case follows}
}

@article{karpukhin2022,
  author = {Karpukhin, Mikhail and M{\'e}tras, Antoine},
  title = {Laplace and {S}teklov extremal metrics via $n$-harmonic maps},
  journal = {Journal of Geometric Analysis},
  volume = {32},
  number = {5},
  pages = {Paper No. 154},
  year = {2022},
  doi = {10.1007/s12220-022-00891-6},
  eprint = {2103.15204},
  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 5.56}
}

Citing this problem

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 It will not change if the problem is solved, reclassified or withdrawn.

OP-01025, "Strict monotonicity of normalised conformal Steklov eigenvalues", clopen.solutions.