Similarity surfaces, connections, and the measurable Riemann mapping theorem

TL;DR

This paper explores a process approximating solutions to the Beltrami equation using similarity surfaces and conformal uniformization, providing new proofs and insights into the analytic dependence of solutions and related geometric structures.

Contribution

It offers an independent proof of the analytic dependence of Beltrami equation solutions and analyzes the convergence of parallel transports related to Schwarz-Christoffel formulas.

Findings

Proves holomorphic dependence of Christoffel symbols without Ahlfors-Bers theorem.

Shows convergence of parallel transports to an affine connection.

Provides theoretical insights into the structure of similarity surfaces and conformal mappings.

Abstract

This article studies a particular process that approximates solutions of the Beltrami equation (straightening of ellipse fields, a.k.a. measurable Riemann mapping theorem) on $\mathbb{C}$. It passes through the introduction of a sequence of similarity surfaces constructed by gluing polygons, and we explain the relation between their conformal uniformization and the Schwarz-Christoffel formula. Numerical aspects, in particular the efficiency of the process, are not studied, but we draw interesting theoretical consequences. First, we give an independent proof of the analytic dependence, on the data (the Beltrami form), of the solution of the Beltrami equation (Ahlfors-Bers theorem). For this we prove, without using the Ahlfors-Bers theorem, the holomorphic dependence, with respect to the polygons, of the Christoffel symbol appearing in the Schwarz-Christoffel formula. Second, these Christoffel symbols define a sequence of parallel transports on the range, and in the case of a data that is $C^2$ with compact support, we prove that it converges to the parallel transport associated to a particular affine connection, which we identify.