Arithmetic Uniformization of Rigid Elliptic Structures: From Rigid to Standard Vekua without the Beltrami Equation
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915897021562880 |
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| author | Alayón-Solarz, Daniel |
| author_facet | Alayón-Solarz, Daniel |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_27210 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Arithmetic Uniformization of Rigid Elliptic Structures: From Rigid to Standard Vekua without the Beltrami Equation Alayón-Solarz, Daniel Complex Variables Analysis of PDEs 30G20 (primary), 30C62, 35J46 (secondary) 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. |
| title | Arithmetic Uniformization of Rigid Elliptic Structures: From Rigid to Standard Vekua without the Beltrami Equation |
| topic | Complex Variables Analysis of PDEs 30G20 (primary), 30C62, 35J46 (secondary) |
| url | https://arxiv.org/abs/2603.27210 |