A sharp two-disk bound for the second positive Neumann eigenvalue under a curvature upper bound
Year: 2026
Type: preprint
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 $Ω\subset\mathbb{C}$ be bounded, simply connected and Lipschitz, with $g=ω|dz|^2$, where $ω$ is continuous and positive on $\overlineΩ$ and $Δ\logω+2Kω\ge0$ in distributions. Write $A=|Ω|_g$ and enumerate the spectrum by $0=λ_0<λ_1\leλ_2\le\cdots$. If $KA<4π$ when $K>0$, then $\frac{1}{λ_2}+\frac{1}{λ_3}>\frac{2}{λ_1(D_K(A/2))}$ 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 $λ_2\le2K$ and $λ_2^{-1}+λ_3^{-1}\ge K^{-1}$. 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.
Source: arXiv (Cornell University)
Authors: Liu, Meiqi, Long, Zhouyu, Zou, Wenming
Institutions: Hangzhou Dianzi University, Tsinghua University
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: 非线性椭圆型方程正规化解和半经典解的渐近性态研究, G3156265683
OpenAlex ID: W7214126831