Sharp upper bounds for Steklov eigenvalues on warped products
Abstract
We obtain sharp isoperimetric-type upper bounds for Steklov eigenvalues on warped products whose bases are compact Riemannian manifolds with boundary and whose fibres are closed Riemannian manifolds. These bounds involve the volume of the warped product and of its boundary, as well as the eigenvalues of the Laplace operator on the fibre and the Steklov eigenvalues of the base. After separation of variables, these bounds are obtained by comparison with auxiliary Helmholtz-Steklov problems on the base involving weighted Laplace operators naturally induced by the warping function. We also study the effect of various natural geometric constraints, such as the warping function being identically equal to one on the boundary. The proofs make use of various spectral approximation methods, including homogenisation on manifolds when the fibres are one-dimensional.
Related
- Sharp bounds for higher mixed Steklov-Robin eigenvalues on domains with holes
- Isodiametric-type upper bounds for Steklov eigenvalues
- Isoperimetric bounds for the weighted Neumann and Steklov eigenvalues in dimensions three through six
- Sharp Bounds for Nodal Sets of Steklov Eigenfunctions on Smooth Surfaces
- Sharp lower bounds for the first eigenvalue of Steklov-type eigenvalue problems on a compact surface
BibTeX
@misc{brisson2026sharp,
title = {{Sharp upper bounds for Steklov eigenvalues on warped products}},
author = {Jade Brisson and Bruno Colbois and Alexandre Girouard and Katie Gittins and Jean Lagacé-Taylor},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13738},
eprint = {2610.13738},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13738},
}
RIS
TY - UNPB
TI - Sharp upper bounds for Steklov eigenvalues on warped products
AU - Jade Brisson
AU - Bruno Colbois
AU - Alexandre Girouard
AU - Katie Gittins
AU - Jean Lagacé-Taylor
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13738
UR - https://arxiv.org/abs/2610.13738
AB - We obtain sharp isoperimetric-type upper bounds for Steklov eigenvalues on warped products whose bases are compact Riemannian manifolds with boundary and whose fibres are closed Riemannian manifolds. These bounds involve the volume of the warped product and of its boundary, as well as the eigenvalues of the Laplace operator on the fibre and the Steklov eigenvalues of the base. After separation of variables, these bounds are obtained by comparison with auxiliary Helmholtz-Steklov problems on the base involving weighted Laplace operators naturally induced by the warping function. We also study the effect of various natural geometric constraints, such as the warping function being identically equal to one on the boundary. The proofs make use of various spectral approximation methods, including homogenisation on manifolds when the fibres are one-dimensional.
ER -