The Burgers Transform: From Holomorphic Functions to Rigid Elliptic Structures

TL;DR

The paper introduces the Burgers transform as a nonlinear bijection linking holomorphic functions to rigid elliptic structures, establishing holomorphicity as necessary for rigidity and exploring its properties through examples.

Contribution

It proves that holomorphicity of the seed function is necessary for rigidity, closing a gap in the literature, and characterizes the transform's domain and properties.

Findings

Holomorphicity is necessary for rigidity of elliptic structures.

The obstruction formula quantifies non-holomorphicity at initial data.

Examples demonstrate the complexity class of seeds and structures are generally unrelated.

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 $\lambda = f(y-\lambda x)$. The output automatically satisfies the conservative complex Burgers equation $\lambda_x+\lambda\lambda_y=0$. 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 $\lambda_x+\lambda\lambda_y=0$ 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. 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.