Sharp harmonic-mean inequalities for Neumann and Aharonov--Bohm spectra

Abstract

We prove sharp two-eigenvalue isoperimetric inequalities for Neumann and Aharonov--Bohm Neumann spectra on surfaces. If $\Omega\subset\mathbb S^2$ is smooth, simply connected and proper, then the harmonic mean of $\mu_2(\Omega)$ and $\mu_3(\Omega)$ is bounded above by the first positive Neumann eigenvalue of the equal-area geodesic disk, with equality only for disks. For simply connected surfaces with Gaussian curvature bounded above, we obtain the magnetic analogue for the first two Aharonov--Bohm eigenvalues. The proof combines a two-dimensional reciprocal Rayleigh--Ritz principle with Green-level comparison. A key additional ingredient is a spectral ordering theorem for magnetic spherical caps: for $0<\nu<1/2$, the first two eigenvalues lie in the angular sectors of effective orders $\nu$ and $1-\nu$. We prove this by the factorization $L_0=T^*T$, $L_1=TT^*$ and an exact Neumann--Dirichlet spectral shift. We also obtain sharp full-sphere and closed-surface bounds, and an annular inequality in terms of conformal modulus and flux.