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
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