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

Daniel Alayón-Solarz

Abstract

For the rigid subclass of variable elliptic structures -- characterized equivalently by the inviscid Burgers law $λ_x+λλ_y=0$ or the self-dilatation $μ_{\bar z}=μμ_z$ -- we show that the auxiliary Beltrami equation in the classical Vekua pipeline is unnecessary. The canonical coordinate $ξ=y-λx$, computed by arithmetic from the spectral parameter $λ$, reduces every rigid variable-algebra Vekua equation to a standard Vekua equation in $ξ$ on any open set where the characteristic Jacobian $Φ=\barξ_x+λ\barξ_y$ does not vanish, with global reduction on domains where $ξ$ is injective. No PDE is solved at any stage.

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 or the self-dilatation -- we show that the auxiliary Beltrami equation in the classical Vekua pipeline is unnecessary. The canonical coordinate , computed by arithmetic from the spectral parameter , reduces every rigid variable-algebra Vekua equation to a standard Vekua equation in on any open set where the characteristic Jacobian does not vanish, with global reduction on domains where is injective. No PDE is solved at any stage.

Paper Structure

This paper contains 33 sections, 20 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. The Vekua workflow and its auxiliary Beltrami equation
  3. Eliminating the auxiliary Beltrami equation
  4. The rigid class: completing the program
  5. Consequences
  6. Plan of the paper
  7. The algebraic picture
  8. The structure polynomial and the moving generator
  9. The Cauchy--Riemann operator and rigidity
  10. Conjugation and norm
  11. The transport picture
  12. The transport map
  13. The universal transport law
  14. Universal intertwining
  15. Rigidity as the Burgers conservation law
  16. ...and 18 more sections

Key Result

Theorem 1.1

Let $(\alpha,\beta)$ be a rigid variable elliptic structure with spectral parameter $\lambda$, and let The reduction is computed by arithmetic from the spectral parameter; no PDE is solved at any stage of the pipeline.

Theorems & Definitions (52)

  • Theorem 1.1: Main theorem, informal
  • Corollary 1.2: Uniformization of the rigid holomorphic class
  • Remark 1.3: Inheritance of classical Vekua theory
  • Example 2.1: Classical complex numbers
  • Example 2.2: Constant elliptic algebras
  • Example 2.3: Genuinely variable structures
  • Definition 2.4: Rigidity
  • Proposition 2.5: Homogeneity
  • Remark 2.6: The 2011 paper and rigidity
  • Proposition 2.7: Leibniz rule
  • ...and 42 more