Steklov isospectral manifolds that are not Neumann isospectral
Two compact Riemannian surfaces with boundary are $\sigma$-isometric if a diffeomorphism between them pulls one metric back to a conformal multiple of the other, with conformal factor equal to $1$ on the boundary; $\sigma$-isometric surfaces have the same Steklov spectrum. The two general techniques for constructing Steklov isospectral manifolds, Sunada's technique and the torus action technique, generally produce manifolds that are also Neumann and Dirichlet isospectral. There is no evident link between the Steklov spectrum and the Neumann or Dirichlet spectrum, so the survey's authors expect an affirmative answer; the same question with Dirichlet in place of Neumann is also open.
Problem. Are there Steklov isospectral connected compact Riemannian manifolds, other than $\sigma$-isometric surfaces, that are not Neumann isospectral?
Source: Posed by Colbois, Girouard, Gordon and Sher (2024), Open Question 9.2.
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, isospectral manifolds, Steklov eigenvalues
References
@article{gordon2021,
author = {Gordon, Carolyn and Herbrich, Peter and Webb, David},
title = {Steklov and {R}obin isospectral manifolds},
journal = {Journal of Spectral Theory},
volume = {11},
number = {1},
pages = {39--61},
year = {2021},
doi = {10.4171/JST/335},
eprint = {1808.10741},
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 9.2}
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01017, "Steklov isospectral manifolds that are not Neumann isospectral", clopen.solutions.