A sharp two-disk bound for the second positive Neumann eigenvalue under a curvature upper bound
Langford and Laugesen conjectured a sharp two-disk bound for the third Neumann eigenvalue under an upper Gaussian-curvature bound (Math. Ann. 386 (2023), 2255--2281, Conjecture 1.4). We prove the conjectured bound for bounded Lipschitz membranes with the weight regularity used by Langford and Laugesen, and strengthen it to a sharp reciprocal inequality. Let $Ω\subset\mathbb{C}$ be a bounded simply connected Lipschitz domain, let $ω\in C^2(Ω)\cap C(\overlineΩ)$ be positive on $\overlineΩ$, and equip $Ω$ with $g=ω|dz|^2$. Suppose $K_g\le K$, $A=\int_Ωω\,dx>0$, and $KA<4π$ when $K>0$. Enumerate the Neumann eigenvalues, counting multiplicity, by $0=λ_0<λ_1\leλ_2\le\cdots$. If $D_K(A/2)$ is the constant-curvature geodesic disk of area $A/2$, then $\frac{1}{λ_2(Ω,g)}+\frac{1}{λ_3(Ω,g)}>\frac{2}{λ_1(D_K(A/2))}$ and $λ_2(Ω,g)<λ_1(D_K(A/2))$. No simplicity of $λ_1$ or boundary differentiability of $ω$ is required. The same conclusions hold for relatively compact disk-type Lipschitz domains in smooth Riemannian surfaces. Both bounds are sharp at fixed area and curvature upper bound: for each admissible $K,A$, a sequence of smooth connected domains of area $A$ in the constant-curvature model has fixed-index Neumann spectra converging to those of $D_K(A/2)\sqcup D_K(A/2)$. Neither extremal value is attained in either connected class. The proof uses two-pole Green coordinates, a positive-kernel comparison, simultaneous centering of two complex moments, and a shifted reciprocal variational estimate.