A sharp two-disk bound for the second positive Neumann eigenvalue under a curvature upper bound
Abstract
We prove the sharp two-disk comparison conjectured by Langford and Laugesen for the second positive Neumann eigenvalue under an upper Gaussian-curvature bound, and strengthen it to a shifted reciprocal inequality. Let Ω⊂ℂ be bounded, simply connected and Lipschitz, with g=ω|dz|2, where ω is continuous and positive on Ω̅ and Δlogω+2Kω≥0 in distributions. Write A=|Ω|g and enumerate the spectrum by 0=λ0<λ1≤λ2≤⋯. If KA<4π when K>0, then 1/λ2+1/λ3>2/λ1(DK(A/2)) and λ2<λ1(DK(A/2)), where DK(A/2) is the constant-curvature disk of area A/2. The same conclusions hold for disk-type Lipschitz domains in smooth surfaces. In the subcritical regime, both bounds are sharp, as witnessed by an extremizing sequence of smooth connected domains of exact area and curvature whose Neumann eigenvalues converge, at every fixed index, to those of two disjoint equal model disks. At KA=4π we obtain the sharp non-strict bounds λ2≤2K and λ2−1+λ3−1≥ K−1. The proof combines two-pole Green coordinates, positive-kernel comparison, and simultaneous centering. We also establish a global quadratic deficit for the model area split.
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BibTeX
@misc{liu2026sharp,
title = {{A sharp two-disk bound for the second positive Neumann eigenvalue under a curvature upper bound}},
author = {Meiqi Liu and Zhouyu Long and Wenming Zou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.26609},
eprint = {2609.26609},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26609},
}
RIS
TY - UNPB
TI - A sharp two-disk bound for the second positive Neumann eigenvalue under a curvature upper bound
AU - Meiqi Liu
AU - Zhouyu Long
AU - Wenming Zou
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2609.26609
UR - https://arxiv.org/abs/2609.26609
AB - We prove the sharp two-disk comparison conjectured by Langford and Laugesen for the second positive Neumann eigenvalue under an upper Gaussian-curvature bound, and strengthen it to a shifted reciprocal inequality. Let Ω⊂ℂ be bounded, simply connected and Lipschitz, with g=ω|dz|^2, where ω is continuous and positive on Ω̅ and Δlogω+2Kω≥0 in distributions. Write A=|Ω|_g and enumerate the spectrum by 0=λ_0<λ_1≤λ_2≤⋯. If KA<4π when K>0, then 1/λ_2+1/λ_3>2/λ_1(D_K(A/2)) and λ_2<λ_1(D_K(A/2)), where D_K(A/2) is the constant-curvature disk of area A/2. The same conclusions hold for disk-type Lipschitz domains in smooth surfaces. In the subcritical regime, both bounds are sharp, as witnessed by an extremizing sequence of smooth connected domains of exact area and curvature whose Neumann eigenvalues converge, at every fixed index, to those of two disjoint equal model disks. At KA=4π we obtain the sharp non-strict bounds λ_2≤2K and λ_2^−1+λ_3^−1≥ K^−1. The proof combines two-pole Green coordinates, positive-kernel comparison, and simultaneous centering. We also establish a global quadratic deficit for the model area split.
ER -