Arithmetic Uniformization of Rigid Elliptic Structures: From Rigid to Standard Vekua without the Beltrami Equation

TL;DR

This paper demonstrates that for a specific class of elliptic structures, the classical Beltrami equation can be bypassed by an arithmetic coordinate transformation, simplifying the analysis of Vekua equations without solving PDEs.

Contribution

It introduces an arithmetic uniformization method that reduces rigid variable-algebra Vekua equations to standard form without using the Beltrami PDE.

Findings

The Beltrami equation is unnecessary for rigid elliptic structures.

A coordinate transformation simplifies Vekua equations.

Global reduction is possible where the coordinate is injective.

Abstract

For the rigid subclass of variable elliptic structures -- characterized equivalently by the inviscid Burgers law $\lambda_x+\lambda\lambda_y=0$ or the self-dilatation $\mu_{\bar z}=\mu\mu_z$ -- we show that the auxiliary Beltrami equation in the classical Vekua pipeline is unnecessary. The canonical coordinate $\xi=y-\lambda x$, computed by arithmetic from the spectral parameter $\lambda$, reduces every rigid variable-algebra Vekua equation to a standard Vekua equation in $\xi$ on any open set where the characteristic Jacobian $\Phi=\bar\xi_x+\lambda\bar\xi_y$ does not vanish, with global reduction on domains where $\xi$ is injective. No PDE is solved at any stage.