Sharp harmonic-mean inequalities for Neumann and Aharonov--Bohm spectra
Abstract
We prove sharp two-eigenvalue isoperimetric inequalities for Neumann and Aharonov--Bohm Neumann spectra on surfaces. If Ω⊂𝕊2 is smooth, simply connected and proper, then the harmonic mean of μ2(Ω) and μ3(Ω) is bounded above by the first positive Neumann eigenvalue of the equal-area geodesic disk, with equality only for disks. For simply connected surfaces with Gaussian curvature bounded above, we obtain the magnetic analogue for the first two Aharonov--Bohm eigenvalues. The proof combines a two-dimensional reciprocal Rayleigh--Ritz principle with Green-level comparison. A key additional ingredient is a spectral ordering theorem for magnetic spherical caps: for 0<ν<1/2, the first two eigenvalues lie in the angular sectors of effective orders ν and 1−ν. We prove this by the factorization L0=T*T, L1=TT* and an exact Neumann--Dirichlet spectral shift. We also obtain sharp full-sphere and closed-surface bounds, and an annular inequality in terms of conformal modulus and flux.
Related
- Sharp Harmonic-Mean Bounds for Neumann Eigenvalues on Riemannian Surfaces and Counterexamples to Geodesic Ball Optimality on Higher-Dimensional Spheres
- Classification of solutions to the Liouville equation with a nonlinear Neumann boundary condition and applications to sharp inequalities
- Sharp Bohr and Bohr-Rogosinski inequalities involving area measure for close-to-convex harmonic mappings
- Berezin--Li--Yau inequalities for Aharonov--Bohm operators
- Spectral Asymptotics of a Singular Harmonic Oscillator with Aharonov--Bohm Flux
BibTeX
@misc{chen2026sharp,
title = {{Sharp harmonic-mean inequalities for Neumann and Aharonov--Bohm spectra}},
author = {Daguang Chen and Chengxi Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24488},
eprint = {2609.24488},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24488},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Sharp harmonic-mean inequalities for Neumann and Aharonov--Bohm spectra
AU - Daguang Chen
AU - Chengxi Yang
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24488
UR - https://arxiv.org/abs/2609.24488
AB - We prove sharp two-eigenvalue isoperimetric inequalities for Neumann and Aharonov--Bohm Neumann spectra on surfaces. If Ω⊂𝕊^2 is smooth, simply connected and proper, then the harmonic mean of μ_2(Ω) and μ_3(Ω) is bounded above by the first positive Neumann eigenvalue of the equal-area geodesic disk, with equality only for disks. For simply connected surfaces with Gaussian curvature bounded above, we obtain the magnetic analogue for the first two Aharonov--Bohm eigenvalues. The proof combines a two-dimensional reciprocal Rayleigh--Ritz principle with Green-level comparison. A key additional ingredient is a spectral ordering theorem for magnetic spherical caps: for 0<ν<1/2, the first two eigenvalues lie in the angular sectors of effective orders ν and 1−ν. We prove this by the factorization L_0=T^*T, L_1=TT^* and an exact Neumann--Dirichlet spectral shift. We also obtain sharp full-sphere and closed-surface bounds, and an annular inequality in terms of conformal modulus and flux.
ER -