Steklov spectra on forms of different degrees for manifolds
Let $\Omega$ be a compact Riemannian manifold with boundary $\Sigma$ and $i \colon \Sigma \to \Omega$ the inclusion. For a $p$-form $\eta$ on $\Sigma$ let $\hat\eta$ be the $p$-form on $\Omega$ with $\Delta\hat\eta = 0$, $i^*\hat\eta = \eta$ and $\nu \lrcorner \hat\eta = 0$. The Raulot–Savo Dirichlet-to-Neumann operator is $\eta \mapsto \nu \lrcorner d\hat\eta$, and $\operatorname{Stek}^{RS}_p(\Omega)$ is its spectrum, the Steklov spectrum when $p = 0$; Karpukhin's Dirichlet-to-Neumann operator on $p$-forms gives a second spectrum $\operatorname{Stek}^K_p(\Omega)$ (for oriented $\Omega$, on co-closed forms); that operator vanishes identically in degree $d = \dim\Sigma$. Quotients $\Gamma \backslash \mathbb B^{d+1}$ of the unit ball by finite orthogonal groups have the same $\operatorname{Stek}^{RS}_p$ exactly when the quotients $\Gamma \backslash \mathbb S^d$ have the same Hodge spectrum on $p$-forms, so pairs of lens spaces of Ikeda and others give flat orbifolds whose Steklov spectra on $p$-forms agree for some but not all $p$. The survey knows of no such pair of manifolds, as opposed to orbifolds. Surfaces related by a diffeomorphism and a conformal factor equal to $1$ on the boundary ($\sigma$-isometric surfaces) have the same Steklov spectrum.
Problem. Are there degrees $p \ne q$ and compact Riemannian manifolds $\Omega_1$ and $\Omega_2$ with boundary, other than $\sigma$-isometric surfaces, such that $\operatorname{Stek}^{RS}_p(\Omega_1) = \operatorname{Stek}^{RS}_p(\Omega_2)$ but $\operatorname{Stek}^{RS}_q(\Omega_1) \ne \operatorname{Stek}^{RS}_q(\Omega_2)$? And the same for Karpukhin's spectra $\operatorname{Stek}^K_p$, with $0 \le p, q \le d - 1$ where $d = \dim \Sigma$?
Source: Posed by Colbois, Girouard, Gordon and Sher (2024), Open Question 9.18.
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, differential forms, isospectral manifolds, Steklov eigenvalues
References
@article{raulot2012,
author = {Raulot, Simon and Savo, Alessandro},
title = {On the first eigenvalue of the {D}irichlet-to-{N}eumann operator on forms},
journal = {Journal of Functional Analysis},
volume = {262},
number = {3},
pages = {889--914},
year = {2012},
doi = {10.1016/j.jfa.2011.10.008}
}
@article{karpukhin2019,
author = {Karpukhin, Mikhail},
title = {The {S}teklov problem on differential forms},
journal = {Canadian Journal of Mathematics},
volume = {71},
number = {2},
pages = {417--435},
year = {2019},
doi = {10.4153/CJM-2018-028-6},
eprint = {1705.08951},
archivePrefix = {arXiv}
}
@incollection{ikeda1989,
author = {Ikeda, Akira},
title = {Riemannian manifolds $p$-isospectral but not $(p+1)$-isospectral},
booktitle = {Geometry of Manifolds (Matsumoto, 1988)},
series = {Perspectives in Mathematics},
volume = {8},
pages = {383--417},
publisher = {Academic Press},
address = {Boston, MA},
year = {1989}
}
@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 9.18}
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01020, "Steklov spectra on forms of different degrees for manifolds", clopen.solutions.