PreprintCC BY 4.0arXiv
Sharp Harmonic-Mean Bounds for Neumann Eigenvalues on Riemannian Surfaces and Counterexamples to Geodesic Ball Optimality on Higher-Dimensional Spheres
Abstract
Let Ω⊂ℂ be a bounded simply connected Lipschitz domain with conformal metric g=ω|dz|2, where ω∈ C2(Ω)∩ L∞(Ω) is positive in the interior and may tend to zero at the boundary. If the Gaussian curvature satisfies Kg≤ K, K>0, and KM<4π, where M=∫Ωω dA, then the Neumann eigenvalues, numbered by 0=μ1(Ω;ω)<μ2(Ω;ω)≤μ3(Ω;ω)≤⋯, satisfy 1/μ2(Ω;ω)+1/μ3(Ω;ω)≥2/μ2(DK(M)). Here DK(M) is the geodesic disk of area M in the surface of constant curvature K. We prove that equality holds exactly when the interiors are isometric. This completes Conjecture 1.2 of Langford--Laugesen; in particular, the model disk maximizes the first positive Neumann eigenvalue in this class.
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BibTeX
@misc{hu2026sharp,
title = {{Sharp Harmonic-Mean Bounds for Neumann Eigenvalues on Riemannian Surfaces and Counterexamples to Geodesic Ball Optimality on Higher-Dimensional Spheres}},
author = {Hongtao Hu and Meiqi Liu and Qiaoran Wu and Wenming Zou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35360},
eprint = {2609.35360},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35360},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sharp Harmonic-Mean Bounds for Neumann Eigenvalues on Riemannian Surfaces and Counterexamples to Geodesic Ball Optimality on Higher-Dimensional Spheres
AU - Hongtao Hu
AU - Meiqi Liu
AU - Qiaoran Wu
AU - Wenming Zou
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35360
UR - https://arxiv.org/abs/2609.35360
AB - Let Ω⊂ℂ be a bounded simply connected Lipschitz domain with conformal metric g=ω|dz|^2, where ω∈ C^2(Ω)∩ L^∞(Ω) is positive in the interior and may tend to zero at the boundary. If the Gaussian curvature satisfies K_g≤ K, K>0, and KM<4π, where M=∫_Ωω dA, then the Neumann eigenvalues, numbered by 0=μ_1(Ω;ω)<μ_2(Ω;ω)≤μ_3(Ω;ω)≤⋯, satisfy 1/μ_2(Ω;ω)+1/μ_3(Ω;ω)≥2/μ_2(D_K(M)). Here D_K(M) is the geodesic disk of area M in the surface of constant curvature K. We prove that equality holds exactly when the interiors are isometric. This completes Conjecture 1.2 of Langford--Laugesen; in particular, the model disk maximizes the first positive Neumann eigenvalue in this class.
ER -