Sharp Harmonic-Mean Bounds for Neumann Eigenvalues on Riemannian Surfaces and Counterexamples to Geodesic Ball Optimality on Higher-Dimensional Spheres

Abstract: Let $Ω\subset\mathbb C$ be a bounded simply connected Lipschitz domain with conformal metric $g=ω|dz|<sup>2$, where $ω\in C^2(Ω)\cap L<sup>\infty(Ω)$ is positive in the interior and may tend to zero at the boundary. If the Gaussian curvature satisfies $K_g\leq K$, $K&gt;0$, and $KM&lt;4π$, where $M=\int_Ωω\,dA$, then the Neumann eigenvalues, numbered by $0=μ_1(Ω;ω)&lt;μ_2(Ω;ω)\leqμ_3(Ω;ω)\leq\cdots$, satisfy $$\frac{1}{μ_2(Ω;ω)}+\frac{1}{μ_3(Ω;ω)}\geq\frac{2}{μ_2(D_K(M))}$$. 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.