The Absorption Theorem for the Beltrami-Vekua Normal Form

Abstract

The Beltrami-Vekua normal form assigns to every smooth first-order real planar elliptic system a complex equation $w_{\bar z}-\mu w_z+\mathcal{A}w+\mathcal{B}\bar w=\mathcal{F}$ by an explicit pipeline. A companion paper showed that the density $\Theta=|\mathcal{B}|^2/(1-|\mu|^2)\,dx\,dy$ and its total mass are invariants under multiplicative gauges $w\mapsto\phi w$ and orientation-preserving diffeomorphisms. The real system carries a larger symmetry: its unknowns may be recombined by any pointwise invertible real-linear substitution $w=\varphi v'+\psi\bar v'$, the complex gauges being the case $\psi\equiv0$. We prove the absorption theorem: re-normalizing through the pipeline after any such substitution returns to the gauge orbit of the original equation, with a universal explicit gauge $\tilde\varphi=-i\lambda/(\varphi-\psi)$, where $\lambda$ is the spectral root of the structure polynomial.