Manifolds of quasiconformal mappings and the nonlinear Beltrami equation

TL;DR

This paper demonstrates that solutions to nonlinear Beltrami equations form a two-dimensional manifold of quasiconformal mappings and that, under certain conditions, this manifold uniquely determines the structure function.

Contribution

It establishes the manifold structure of solutions to nonlinear Beltrami equations and proves the uniqueness of the structure function under regularity assumptions.

Findings

Solutions form a 2D manifold of quasiconformal mappings

The manifold structure is explicitly characterized

The structure function is uniquely determined by the manifold

Abstract

In this paper we show that the homeomorphic solutions to each nonlinear Beltrami equation $\partial_{\bar{z}} f = \mathcal{H}(z, \partial_{z} f)$ generate a two-dimensional manifold of quasiconformal mappings $\mathcal{F}_{\mathcal{H}} \subset W^{1,2}_{\mathrm{loc}}(\mathbb{C})$. Moreover, we show that under regularity assumptions on $\mathcal{H}$, the manifold $\mathcal{F}_{\mathcal{H}}$ defines the structure function $\mathcal{H}$ uniquely.