Variable Elliptic Structures on the Plane: Transport Dynamics, Rigidity, and Function Theory

Abstract

We develop a theory of variable elliptic structures on planar domains, in which the imaginary unit $i(x,y)$ is a moving generator of a rank-two real algebra bundle defined by a smoothly varying quadratic relation. Differentiating this relation produces an intrinsic obstruction $G = i_x + i\, i_y$ that governs all deviations from the constant-coefficient theory, such as the inhomogeneity of the generalized Cauchy-Riemann system and the forcing of a universal complex inviscid Burgers equation satisfied by the spectral parameter. The vanishing of $G$ -- rigidity -- selects the conservative regime of this transport law and simultaneously restores a coherent function theory: Cauchy-Pompeiu representation, covariant holomorphicity with gauge structure, a similarity principle, and a factorization of the variable Laplacian. A rigidity-flatness theorem shows that the only structure that is both rigid and Riemannian-flat is the constant one. Translated into Beltrami coordinates, the rigidity condition becomes $\mu_{\bar{z}} = \mu\, \mu_z$: the structure map satisfies its own Beltrami equation, a self-dilatation property in the Poincar\'e disk. The central result is the Fundamental Independence Theorem: the Beltrami modulus $\|\mu\|_{C^0}$ (zeroth order) and the transport obstruction $\|R(\mu)\|_{C^{0,\alpha}}$ (first order) are independently prescribable.