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

Let $Ω\subset\mathbb C$ be a bounded simply connected Lipschitz domain with conformal metric $g=ω|dz|^2$, where $ω\in C^2(Ω)\cap L^\infty(Ω)$ is positive in the interior and may tend to zero at the boundary. If the Gaussian curvature satisfies $K_g\leq K$, $K>0$, and $KM<4π$, where $M=\int_Ωω\,dA$,