Curvature properties determined by the Steklov spectrum in dimension three and higher
Let $\Omega$ be a compact Riemannian manifold with boundary, of dimension at least $3$. The torus action technique, applied to examples of Gordon and Szabó, gives pairs of manifolds that are Steklov, Neumann and Dirichlet isospectral, with Laplace isospectral boundaries, such that one manifold has constant Ricci curvature and the other not, or one boundary has constant scalar curvature and the other not, or one manifold has parallel curvature tensor and the other not. The survey notes that the questions below are open for the Laplacian too, where it is known only that a metric of constant curvature on a closed manifold is spectrally isolated (Tanno, Kuwabara, Sharafutdinov).
Problem. Can one tell from the Steklov spectrum of $\Omega$ whether (a) $\Omega$ has constant sectional curvature; (b) the metric induced on the boundary is Einstein; (c) the metric induced on the boundary has constant sectional curvature; (d) the mean curvature of the boundary is constant; (e) the principal curvatures of the boundary are constant?
Source: Posed by Colbois, Girouard, Gordon and Sher (2024), Open Question 9.15.
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, spectral invariants, Steklov eigenvalues
References
@article{gordon2002,
author = {Gordon, Carolyn and Szab{\'o}, Zolt{\'a}n I.},
title = {Isospectral deformations of negatively curved {R}iemannian manifolds with boundary which are not locally isometric},
journal = {Duke Mathematical Journal},
volume = {113},
number = {2},
pages = {355--383},
year = {2002},
doi = {10.1215/S0012-7094-02-11326-X}
}
@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 9.15}
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01019, "Curvature properties determined by the Steklov spectrum in dimension three and higher", clopen.solutions.