Moduli of the Sourceless Framed Beltrami-Vekua Normal Form

Abstract

We study the moduli of sourceless framed Beltrami-Vekua equations under recombinations of the unknown, scalings, and orientation-preserving changes of variables. On every bounded simply connected domain, such an equation reduces to $w_{\bar z}=B\bar w$ on the unit disk, with residual symmetries given exactly by zero-free holomorphic gauges and Möbius transformations. When $B$ is zero-free, the equation is completely classified by two data modulo Möbius: the hyperbolic mass density $\vartheta=\tfrac14(1-|z|^2)^2|B|^2$ and the phase-curvature current $K=Δ\arg B$, whose hyperbolic density at $C^2$ regularity is $κ=Δ_{\mathrm{hyp}}\arg B$. We determine the exact range of these invariants: every positive Hölder density and every phase current arising as the Laplacian of a Hölder phase occur. For fields with zeros, the classification extends to the phase-integrable sector through the triple $(\vartheta,dη,d{\star}η)$, where $η=\operatorname{Im}(dB/B)$. On the tame sector, $dη$ is the atomic charge measure, while $d{\star}η$ carries the remaining phase curvature. Thus the pseudo-analytic mass and charge are numerical projections of a larger infinite-dimensional moduli space. Explicit equal-mass, equal-charge, inequivalent equations are exhibited.