Does the Steklov spectrum detect interior singularities of orbifolds?
The Dirichlet-to-Neumann operator and the Steklov spectrum of a compact Riemannian orbifold with boundary were introduced by Arias-Marco, Dryden, Gordon, Hassannezhad, Ray and Stanhope. In dimension two, interior cone points and dihedral points can be removed by a conformal change of the metric equal to $1$ on the boundary, which does not change the Steklov spectrum, so the spectrum does not detect them; whether it detects reflectors that do not meet the boundary is unknown. In dimension greater than two the question is open.
Problem. Does the Steklov spectrum of a compact Riemannian orbifold of dimension at least three detect the presence of singularities in its interior?
Source: Posed by Colbois, Girouard, Gordon and Sher (2024), Open Question 8.18, after Arias-Marco, Dryden, Gordon, Hassannezhad, Ray and Stanhope (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, inverse spectral problems, orbifolds, Steklov eigenvalues
References
@article{ariasmarco2019,
author = {Arias-Marco, Teresa and Dryden, Emily B. and Gordon, Carolyn S. and Hassannezhad, Asma and Ray, Allie and Stanhope, Elizabeth},
title = {Spectral geometry of the {S}teklov problem on orbifolds},
journal = {International Mathematics Research Notices},
number = {1},
pages = {90--139},
year = {2019},
doi = {10.1093/imrn/rnx117},
eprint = {1609.05142},
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 8.18}
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01014, "Does the Steklov spectrum detect interior singularities of orbifolds?", clopen.solutions.