The Theory of Quasiconformal Mappings in Higher Dimensions, I

We present a survey of the many and various elements of the modern higher-dimensional theory of quasiconformal mappings and their wide and varied application. It is unified (and limited) by the theme of the author's interests. Thus we will discuss the basic theory as it developed in the 1960s in the early work of F.W. Gehring and Yu G. Reshetnyak and subsequently explore the connections with geometric function theory, nonlinear partial differential equations, differential and geometric topology and dynamics as they ensued over the following decades. We give few proofs as we try to outline the major results of the area and current research themes. We do not strive to present these results in maximal generality, as to achieve this considerable technical knowledge would be necessary of the reader. We have tried to give a feel of where the area is, what are the central ideas and problems and where are the major current interactions with researchers in other areas. We have also added a bit of history here and there. We have not been able to cover the many recent advances generalising the theory to mappings of finite distortion and to degenerate elliptic Beltrami systems which connects the theory closely with the calculus of variations and nonlinear elasticity, nonlinear Hodge theory and related areas, although the reader may see shadows of this aspect in parts

Stream lines, quasilines and holomorphic motions

We give a new application of the theory of holomorphic motions to the study the distortion of level lines of harmonic functions and stream lines of ideal planar fluid flow. In various settings, we show they are in fact quasilines - the quasiconformal images of the real line. These methods also provide quite explicit global estimates on the geometry of these curves.Comment: 10 pages, 3 figure

Super regularity for Beltrami systems

We prove a surprising higher regularity for solutions to the nonlinear elliptic autonomous Beltrami equation in a planar domain $\Omega$, f_\overline{z} = \mathcal{A}(f_z)\ a.e.\ z\in\Omega, when $\mathcal{A}$ is linear at $\infty$. Namely $W^{1,1}_{\operatorname{loc}}(\Omega)$ solutions are $W^{2,2+\epsilon}_{\operatorname{loc}}(\Omega)$. Here \epsilon>0 depends explicitly on the ellipticity bounds of $\mathcal{A}$. The condition "is linear at $\infty$" is necessary - the result is false for the equation f_\overline{z} = k|f_z|, for any 0<k<1, ($k=0$ is Weyl's lemma) and the improved regularity is sharp, but can be further improved if, for instance, $\mathcal{A}$ is smooth. We also discuss the subsequent higher regularity implications for fully non-linear Beltrami systems f_\overline{z} = \mathcal{A}(z, f_z)\ a.e.\ z\in\Omega. There the condition "linear at $\infty$" also implies improved regularity for $W^{1,1}_{\operatorname{loc}}(\Omega)$ solutions.