Large normalised first Steklov eigenvalue with the boundary metric fixed
Let $\Omega$ be a compact connected manifold of dimension $d + 1 \ge 3$ with boundary $\Sigma$, and for a Riemannian metric $g$ on $\Omega$ let $\sigma_1(\Omega, g)$ be its first nonzero Steklov eigenvalue. Colbois, El Soufi and Girouard constructed metrics conformal to a given one and equal to it on $\Sigma$ with $\sigma_1$ arbitrarily large; within a conformal class, however, the normalised eigenvalue $\sigma_1(\Omega, g)\, |\Sigma|_g\, |\Omega|_g^{(1-d)/(d+1)}$ stays bounded. Cianci and Girouard made it arbitrarily large, with the metric fixed on $\Sigma$ and $|\Omega|_g = 1$, when $d + 1 \ge 4$, $\Sigma$ is connected and its metric admits a unit Killing vector field whose dual $1$-form has nowhere-vanishing exterior derivative, as on odd-dimensional round spheres. If only the volume of a connected $\Sigma$ is prescribed and $d + 1 \ge 4$, a cylindrical collar $\Sigma \times [0, 1]$ whose cross-sections have unit volume and large first Laplace eigenvalue gives such examples.
Problem. Let $(\Omega, g)$ be a compact connected Riemannian manifold of dimension $d + 1 \ge 3$ with boundary $\Sigma$. Is there a family $(g_\varepsilon)_{\varepsilon > 0}$ of Riemannian metrics on $\Omega$, all equal to $g$ on $\Sigma$, such that $$\sigma_1(\Omega, g_\varepsilon)\, |\Sigma|_{g_\varepsilon}\, |\Omega|_{g_\varepsilon}^{(1-d)/(d+1)} \to \infty \quad \text{as } \varepsilon \to 0?$$
Source: Posed by Colbois, Girouard, Gordon and Sher (2024), Open Question 4.14, after Cianci and Girouard (2018).
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, eigenvalue bounds, prescribed boundary, Steklov eigenvalues
References
@article{cianci2018,
author = {Cianci, Donato and Girouard, Alexandre},
title = {Large spectral gaps for {S}teklov eigenvalues under volume constraints and under localized conformal deformations},
journal = {Annals of Global Analysis and Geometry},
volume = {54},
number = {4},
pages = {529--539},
year = {2018},
doi = {10.1007/s10455-018-9612-6}
}
@article{colbois2019,
author = {Colbois, Bruno and El Soufi, Ahmad and Girouard, Alexandre},
title = {Compact manifolds with fixed boundary and large {S}teklov eigenvalues},
journal = {Proceedings of the American Mathematical Society},
volume = {147},
number = {9},
pages = {3813--3827},
year = {2019}
}
@misc{girouard2024,
author = {Girouard, Alexandre and Polymerakis, Panagiotis},
title = {Large {S}teklov eigenvalues under volume constraints},
year = {2024},
eprint = {2403.08925},
archivePrefix = {arXiv},
note = {Theorem 3: the answer is yes when the boundary is connected and isometric to a Riemannian product of closed manifolds}
}
@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 4.14}
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01022, "Large normalised first Steklov eigenvalue with the boundary metric fixed", clopen.solutions.