The Burgers Transform: From Holomorphic Functions to Rigid Elliptic Structures
Daniel Alayón-Solarz
TL;DR
The paper establishes the Burgers transform $\mathcal{B}$ as a bijection from holomorphic seeds $f:U\to\mathbb{C}_+$ to rigid variable elliptic plane structures, and proves holomorphicity of $f$ is a necessary condition for rigidity via an obstruction $H|_{x=0}=2i\,\operatorname{Im}f\,f_{\bar w}$. It develops the $f$-analytics hierarchy, showing holomorphic seeds are the maximal seed class compatible with rigidity and analysing domain formation through shocks, the seed Jacobian, and affine-equivariance. The geometry is framed through Beltrami leaves and a twisted affine action, with a precise propagator $\mathcal{P}_f[h](x,y)=\frac{h(w_0)}{J_f}$ that yields a Jacobian-twisted multiplicativity $\mathcal{P}_f[h_1 h_2]=J_f\,\mathcal{P}_f[h_1]\mathcal{P}_f[h_2]$ and a deformed product on outputs. A rich suite of worked examples ($\varepsilon$-, $\delta$-, exponential, and Cauchy-kernel seeds) demonstrates how seeds with different analytic/complex profiles generate diverse rigid structures, including globally defined domains (exponential seed) and mixed boundary phenomena (epsilon/delta families). The work culminates in a framework linking self-reference, holomorphic seeds, and rigidity, with numerous open problems on algebraic structure, global domains, and higher-dimensional generalisations, highlighting the deep role of holomorphicity in emergent complex rigidity.
Abstract
We introduce the Burgers transform $\mathcal{B}$, a nonlinear bijection between holomorphic functions $f\colon U\to\mathbb{C}^+$ and rigid variable elliptic structures on the plane, defined implicitly by $λ= f(y-λx)$. The output automatically satisfies the conservative complex Burgers equation $λ_x+λλ_y=0$. Our main result is that holomorphicity of the seed $f$ is necessary, not merely sufficient, for rigidity: any $C^1$ function whose implicit solution satisfies $λ_x+λλ_y=0$ must be holomorphic. This closes a gap in the existing literature and identifies $\operatorname{Hol}(U,\mathbb{C}^+)$ as the maximal seed space compatible with rigidity. The obstruction formula $H|_{x=0} = 2i\,(\operatorname{Im} f)\,f_{\bar{w}}$ quantifies the cost of non-holomorphicity at the level of the initial data. The transform establishes a hierarchy we call $f$-analytics: standard complex analysis ($f=i$), $p(x)$-analytics ($f\colon U\to i\mathbb{R}_+$), and the full rigid class ($f\colon U\to\mathbb{C}^+$ holomorphic), in which rigidity -- absent from the real diameter of $\mathbb{D}$ -- emerges as a genuinely complex phenomenon. We characterise the domain of $\mathcal{B}$ through shock formation, its interaction with affine automorphisms of $\mathbb{C}^+$, and the infinitesimal structure: the propagator $\mathcal{P}_f = D\mathcal{B}_f$ satisfies a Jacobian-twisted multiplicativity that deforms the seed algebra by the density of characteristics. Four worked examples -- affine, exponential, inverse, and trigonometric seeds -- show that the complexity class of a seed and that of the resulting structure are generically unrelated.