The Pseudo-Analytic Mass of a Beltrami-Vekua Equation

Abstract: Every smooth first-order real planar elliptic system admits a universal complex form $w_{\bar z} - μw_z + \mathcal{A} w + \mathcal{B} \bar w = \mathcal{F}$, which we call the Beltrami-Vekua equation: the data $(μ, \mathcal{A}, \mathcal{B}, \mathcal{F})$ are produced from the original system by algebraic operations and differentiations, with no auxiliary PDE. On this space we study the joint action of multiplicative gauges $w \mapsto φw$ and orientation-preserving diffeomorphisms. Our main result is that the 2-form $Θ= |\mathcal{B}|^2 / (1 - |μ|^2) \, dx \, dy$ is gauge-invariant and pulls back covariantly under diffeomorphisms; its form is forced, with $|\mathcal{B}|^2$ the unique $\mathcal{B}$-quadratic combination invariant under $\mathcal{B} \mapsto \mathcal{B}φ/\barφ$ and $1 - |μ|^2$ the conformal distortion factor from the diffeomorphism law for $μ$. The total mass $\mathcal{M}(D) = \int_ΩΘ$, the \emph{pseudo-analytic mass}, vanishes precisely on the analytic class $\mathcal{B} \equiv 0$ and separates a continuous family of pairwise inequivalent pseudo-analytic equations on the disk. As a by-product, Vekua's two-stage reduction - uniformization then gauge elimination - requires only one variable-coefficient PDE solve: the Beltrami diffeomorphism supplies the integrating factor for a flat $\bar\partial$-equation.