Prescribed Steklov spectra and multiplicities on surfaces
Abstract
Let 0lt a1≤⋯≤ aN and Hgt aN. On every compact connected orientable surface with b≥1 boundary components and genus at least an explicit integer Γ(N,b), we construct a smooth metric whose first N positive Steklov eigenvalues are a1,…,aN and whose next eigenvalue exceeds H. Repetitions are allowed. For connected boundary, Γ(N,1)=⌈ N(N−1)/4⌉. This is the least genus permitted by the complete-graph band construction used in the proof; no optimality among all metrics is asserted. The metric may also have any sufficiently small prescribed boundary length, any positive prescribed area, and trivial isometry group. All these statements concern the ordinary, unweighted Steklov problem. The proof realises positive graph Laplacians by Dirichlet and boundary forms, estimates the complete finite spectral matrix, and applies degree after smoothing. For a family of invariant perforations of a closed hyperbolic surface, we identify the Steklov spectral subspace converging to the first positive Laplace eigenvalue with its Laplace eigenspace, equivariantly under the finite group used in the construction. The first Steklov multiplicity is at least the smallest dimension of a real irreducible constituent of that Laplace eigenspace; if the latter is irreducible, its multiplicity is preserved exactly. The surfaces of Fortier Bourque, Gruda-Mediavilla, Petri and Pineault yield hyperbolic examples of genera $10$, $17$ and $37$ with first Steklov multiplicities $16$, $21$ and at least $24$, respectively. They disprove Jammes's proposed equality between maximal first multiplicity and the relative chromatic number minus one.
Related
- Exponential Growth of Mean Multiplicities in Length Spectra of Semi-Arithmetic Surfaces of Arbitrary Arithmetic Dimension
- Nearest structured matrix having an eigenvalue with prescribed lower bounds of algebraic and geometric multiplicities
- Discreteness of the Steklov Spectrum for Exterior Non-Compact Free-Boundary Minimal Surfaces with Regular Ends of Finite Total Curvature
- Sharp Bounds for Nodal Sets of Steklov Eigenfunctions on Smooth Surfaces
- Prescribed mean curvature surfaces of Riemann minimal type
BibTeX
@misc{nifa2026prescribed,
title = {{Prescribed Steklov spectra and multiplicities on surfaces}},
author = {Anass Nifa},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.5281/zenodo.22949635},
url = {https://hal.science/hal-05769929v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Prescribed Steklov spectra and multiplicities on surfaces
AU - Anass Nifa
PY - 2026
DA - 2026/09/04
PB - HAL
DO - 10.5281/zenodo.22949635
UR - https://hal.science/hal-05769929v1
AB - Let 0lt a_1≤⋯≤ a_N and Hgt a_N. On every compact connected orientable surface with b≥1 boundary components and genus at least an explicit integer Γ(N,b), we construct a smooth metric whose first N positive Steklov eigenvalues are a_1,…,a_N and whose next eigenvalue exceeds H. Repetitions are allowed. For connected boundary, Γ(N,1)=⌈ N(N−1)/4⌉. This is the least genus permitted by the complete-graph band construction used in the proof; no optimality among all metrics is asserted. The metric may also have any sufficiently small prescribed boundary length, any positive prescribed area, and trivial isometry group. All these statements concern the ordinary, unweighted Steklov problem. The proof realises positive graph Laplacians by Dirichlet and boundary forms, estimates the complete finite spectral matrix, and applies degree after smoothing. For a family of invariant perforations of a closed hyperbolic surface, we identify the Steklov spectral subspace converging to the first positive Laplace eigenvalue with its Laplace eigenspace, equivariantly under the finite group used in the construction. The first Steklov multiplicity is at least the smallest dimension of a real irreducible constituent of that Laplace eigenspace; if the latter is irreducible, its multiplicity is preserved exactly. The surfaces of Fortier Bourque, Gruda-Mediavilla, Petri and Pineault yield hyperbolic examples of genera $10$, $17$ and $37$ with first Steklov multiplicities $16$, $21$ and at least $24$, respectively. They disprove Jammes's proposed equality between maximal first multiplicity and the relative chromatic number minus one.
ER -