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Similarity surfaces, connections, and the measurable Riemann mapping theorem

Dec 2022

This article studies a particular process that approximates solutions of theBeltrami equation (straightening of ellipse fields, a.k.a. measurable Riemannmapping theorem) on \$\mathbb{C}\$. It passes through the introduction of asequence of similarity surfaces constructed by gluing polygons, and we explainthe relation between their conformal uniformization and the Schwarz-Christoffelformula. Numerical aspects, in particular the efficiency of the process, arenot studied, but we draw interesting theoretical consequences. First, we givean independent proof of the analytic dependence, on the data (the Beltramiform), of the solution of the Beltrami equation (Ahlfors-Bers theorem). Forthis we prove, without using the Ahlfors-Bers theorem, the holomorphicdependence, with respect to the polygons, of the Christoffel symbol appearingin the Schwarz-Christoffel formula. Second, these Christoffel symbols define asequence 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 paralleltransport associated to a particular affine connection, which we identify.

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