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

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.

arXiv comment17 pages. Comments and corrections are welcome