Equivariant Steklov homogenisation and stability of spectral representations
Abstract
We prove that isolated spectral clusters preserve their finite-group representation under equivariant spectral convergence. In particular, an irreducible limiting eigenspace cannot split into several nearby eigenvalues, while an isotypic limiting eigenspace imposes corresponding divisibility constraints on the multiplicities in the approximating cluster. We then construct a one-parameter equivariant Steklov homogenisation on closed orientable surfaces, allowing Laplace spectral representations to be transferred to Steklov spectral clusters. Applied to the hyperbolic surfaces of Fortier Bourque, Gruda-Mediavilla, Petri, and Pineault, this yields first Steklov multiplicities 16 and 21 in genera 10 and 17, and at least 24 in genus 37. These examples give counterexamples to Jammes's conjecture on first Steklov multiplicity.
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BibTeX
@misc{audetbeaumont2026equivariant,
title = {{Equivariant Steklov homogenisation and stability of spectral representations}},
author = {Samuel Audet-Beaumont and Maxime Fortier Bourque and Alexandre Girouard},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12725},
eprint = {2610.12725},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12725},
}
RIS
TY - UNPB
TI - Equivariant Steklov homogenisation and stability of spectral representations
AU - Samuel Audet-Beaumont
AU - Maxime Fortier Bourque
AU - Alexandre Girouard
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12725
UR - https://arxiv.org/abs/2610.12725
AB - We prove that isolated spectral clusters preserve their finite-group representation under equivariant spectral convergence. In particular, an irreducible limiting eigenspace cannot split into several nearby eigenvalues, while an isotypic limiting eigenspace imposes corresponding divisibility constraints on the multiplicities in the approximating cluster. We then construct a one-parameter equivariant Steklov homogenisation on closed orientable surfaces, allowing Laplace spectral representations to be transferred to Steklov spectral clusters. Applied to the hyperbolic surfaces of Fortier Bourque, Gruda-Mediavilla, Petri, and Pineault, this yields first Steklov multiplicities 16 and 21 in genera 10 and 17, and at least 24 in genus 37. These examples give counterexamples to Jammes's conjecture on first Steklov multiplicity.
ER -