Does the Steklov spectrum determine the Laplace spectrum of the boundary?
Two compact Riemannian manifolds with boundary are Steklov isospectral if their Dirichlet-to-Neumann operators have the same eigenvalues with multiplicity. The Steklov spectrum of a surface determines the lengths of its boundary components (Girouard, Parnovski, Polterovich and Sher), hence the Laplace spectrum of its boundary. In all examples known to the survey ($\sigma$-isometric surfaces, Sunada's technique, the torus action technique, products, quotients of balls) the boundaries are Laplace isospectral as well.
Problem. Are there Steklov isospectral compact Riemannian manifolds whose boundaries are not Laplace isospectral? Equivalently, does the Steklov spectrum of a compact Riemannian manifold determine the Laplace spectrum of its boundary?
Source: Girouard, Parnovski, Polterovich and Sher, The Steklov spectrum of surfaces: asymptotics and invariants, Math. Proc. Cambridge Philos. Soc. 157 (2014), Question 1.11; Girouard and Polterovich (2017), Open Problem 7. Listed as Open Question 9.1 in Colbois–Girouard–Gordon–Sher (2024).
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, inverse spectral problems, isospectral manifolds, Steklov eigenvalues
References
@article{girouard2014,
author = {Girouard, Alexandre and Parnovski, Leonid and Polterovich, Iosif and Sher, David A.},
title = {The {S}teklov spectrum of surfaces: asymptotics and invariants},
journal = {Mathematical Proceedings of the Cambridge Philosophical Society},
volume = {157},
number = {3},
pages = {379--389},
year = {2014},
doi = {10.1017/S030500411400036X},
eprint = {1311.5533},
archivePrefix = {arXiv}
}
@article{girouard2017,
author = {Girouard, Alexandre and Polterovich, Iosif},
title = {Spectral geometry of the {S}teklov problem},
journal = {Journal of Spectral Theory},
volume = {7},
number = {2},
pages = {321--359},
year = {2017},
doi = {10.4171/JST/164},
eprint = {1411.6567},
archivePrefix = {arXiv}
}
@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}
}
@misc{florentin2025,
author = {Florentin, Benjamin},
title = {Steklov isospectrality of conformal metrics},
year = {2025},
eprint = {2501.15535},
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 9.1}
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01016, "Does the Steklov spectrum determine the Laplace spectrum of the boundary?", clopen.solutions.