Introduction And Overview

Abstract

Abstract The interplay between partial differential equations (PDEs) and the theory of mappings has a long and distinguished history, and that connection underpins this book. Gauss’s practical geodesic survey work stimulated him to develop the theory of conformal transformations, for mapping figures from one surface to another. For conformal transformation from plane to plane he used a pair of equations apparently derived by d’ Alembert, who first related the derivatives of the real and imaginary part of a complex function in 1746 in his work on hydrodynamics (31], p. 497. These equations have become known as the Cauchy–Riemann equations. Gauss developed the differential geometry of surfaces around 1827, emphasizing the intrinsic geometry, with Gaussian curvature defined by measurements within the surface. If a surface is deformed conformally (preserving angles), then the Gaussian curvature is unchanged, and hence the intrinsic geometry of the surface is unaffected by such deformations. Gauss also considered geodesic curves within surfaces. In 1829 Lobachevsky constructed a surface (the horosphere) within his non-Euclidean space, such that the intrinsic geometry within that surface is Euclidean, with geodesic curves being called Euclidean lines. For the converse process, he could only suggest tentatively that, within Euclidean space, the intrinsic geometry of a sphere of imaginary radius was Lobachevskian. But imaginary numbers were then regarded with justifiable suspicion, and he did not propose that as an acceptable model of his geometry within Euclidean space. In his most famous work, Beltrami [32] showed that Lobachevsky’s geometry is the intrinsic geometry of a surface of constant negative curvature, with geodesic curves being called lines in Lobachevsky’s geometry. Beltrami illustrated various surfaces with constant negative curvature, the simplest of which is the pseudosphere generated by revolving a tractrix around its axis. Beltrami’s paper convinced most mathematicians that the geometries of Euclid and of Lobachevsky are logically equivalent. In that work Beltrami used a differential equation corresponding to Gauss’s equation. This has come to be known as Beltrami’s equation, and later in this book we shall present the most recent developments in this area, solving Beltrami’s equation at the critical point. where uniform ellipticity bounds are lost. This will necessitate the development of some considerable technical machinery to enable us to move away from the classical setting of uniformly elliptic PDEs to the case of degenerate elliptic equations. Beltrami’s equation and its solutions, the quasiconformal mappings, have found a home in virtually all aspects of modern complex analysis, from the theory of Riemann surfaces and Teichmüller and Moduli spaces to more recent developments such as holomorphic dynamics and three-dimensional hyperbolic geometry. We hope the developments presented in this book encourage new applications of quasiconformal mappings in these areas.