CHAPTER 1

Introduction

The interplay between Partial Differential Equations (PDEs) and the theory of

mappings has a long and distinguished history. Gauss' practical work of geodesic

surveying 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 [15] pg 497. These equations have become

known as the Cauchy-Riemann equations. Gauss developed the differential geome-

try of surfaces (1827) emphasizing the intrinsic geometry, with Gaussian curvature

defined by measurements within the surface. 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. The most famous work of Beltrami

is [16] from 1867. There he showed that Lobachevsky's geometry is the intrinsic

geometry of a surface of constant negative curvature, with geodesic curves being

called lines in this geometry. Beltrami illustrated various surfaces with constant

negative curvature, the simplest of which is the pseudosphere generated by revolv-

ing 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 the Beltrami equation. This major work of Beltrami's

appeared 3 years after he took up the geodesy professorship at Pisa after it was

turned down by Riemann. Beltrami went on to become one of Italy's foremost

mathematicians, primarily as a geometer but making significant contributions in

analysis, the theory of heat, and to mechanics.

Of course since those early times, the theory of conformal mappings, analytic

functions, Riemann surfaces and so forth has expanded in many different directions,

far too numerous to relate, and now lie at the foundation of much of modern

mathematics. Moreover practical applications, such as in fluid flow, hydrodynamics

and more modern areas of control theory and robotics etc, abound.

In this paper we present the most recent developments concerning certain linear

and non-linear equations in the plane, solving Beltrami's equation at the critical

point, where the uniform eUipticity bounds are lost. These provide far reaching

l