The Pseudo-Analytic Mass of a Beltrami-Vekua Equation

Abstract

Every smooth first-order real planar elliptic system admits a universal complex form $w_{\bar{z}}-\mu w_z+\mathcal{A}w+\mathcal{B}\bar{w}=\mathcal{F}$, which we call the Beltrami-Vekua equation: the data $(\mu ,\mathcal{A},\mathcal{B},\mathcal{F})$ 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 $w\mapsto \varphi w$ and orientation-preserving diffeomorphisms. Our main result is that the 2-form $\Theta =|\mathcal{B}{|}^2/(1-|\mu {|}^2)dxdy$ 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 $\mathcal{B}\mapsto \mathcal{B}\varphi /\bar{\varphi }$ and $1-|\mu {|}^2$ the conformal distortion factor from the diffeomorphism law for $\mu$. The total mass $\mathcal{M}(D)={\int }_{\Omega }\Theta$, 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.

20 pages