Steklov isospectral manifolds that are not strongly Steklov isospectral
For a compact Riemannian manifold $\Omega$ with boundary and a real $\alpha$ that is not a Dirichlet eigenvalue of $\Omega$, let $\operatorname{Stek}_\alpha(\Omega)$ be the spectrum of the problem $\Delta u = \alpha u$ in $\Omega$, $\partial_\nu u = \sigma u$ on $\partial\Omega$; so $\operatorname{Stek}_0(\Omega)$ is the Steklov spectrum. Two such manifolds are strongly Steklov isospectral if they have the same Dirichlet spectrum and the same $\operatorname{Stek}_\alpha$ for every $\alpha$ outside it; equivalently, they are isospectral for the Robin problem with every parameter. Apart from $\sigma$-isometric surfaces, all Steklov isospectral manifolds known to the survey are strongly Steklov isospectral. For $\alpha \ne 0$ the spectrum $\operatorname{Stek}_\alpha$ of a surface is not invariant under $\sigma$-isometries (diffeomorphisms pulling one metric back to a conformal multiple of the other, with conformal factor equal to $1$ on the boundary), which preserve the Steklov spectrum.
Problem. Are there Steklov isospectral connected compact Riemannian manifolds, other than $\sigma$-isometric surfaces, that are not strongly Steklov isospectral?
Source: Posed by Colbois, Girouard, Gordon and Sher (2024), Open Question 9.5.
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, Robin problem, 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 9.5}
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01018, "Steklov isospectral manifolds that are not strongly Steklov isospectral", clopen.solutions.