One of the central results in the subject of complex analysis, the Riemann mappingtheorem unifies the areas of algebra, geometry, and topology. Today it is recognized as one of the most important theorems of nineteenth-century mathematics, even though its original proof by Georg Freidrich Bernhard Riemann was flawed. The search for generalizations and a better understanding of this theorem has continued throughout the twentieth century.
Early mapmakers, from Hipparchus in ancient Greece to Gerhardus Mercator in 16th-century Flanders, discovered that certain types of maps of the globe preserve shape information on a small scale. This means that the map accurately represents small circles as circles (rather than as ellipses or ovals), and that it accurately represents angles. Such maps are useful for navigation, even though they may grossly distort large-scale features. (For example, Greenland is disproportionately large in the Mercator projection.) A locally shape-preserving map is called conformal, and there are many conformal projections besides Mercator's.
A truly systematic theory of conformal mappings became possible in the early 1800s, when mathematicians discovered these maps can be represented by complex analytic functions. Any point in the region to be mapped can be labeled by coordinates (x, y), and these coordinates can be transformed into a complex number x + iy. Similarly, any point in the image of this region can be described by coordinates (x', y'), which can be thought of a complex number x' + iy'. The function that assigns each point x + iy to its image x' + iy' turns out to be differentiable, which means that all the tools of calculus can be applied to it. However, it is not merely differentiable as a function of real numbers; it is also differentiable as a function of complex numbers. The extra structure of complex numbers (specifically, the fact that they constitute a field rather than merely a vector space) has a profound effect in the theory of such functions, which are therefore called "analytic" rather than "differentiable."
For many years, complex analysts perfected laborious formulas for mapping one particular type of planar region, such as the interior of a triangle, to another by means of analytic (conformal) functions. These formulas were used by engineers and physicists, for example to draw the equipotential curves of a distribution of electric charge. However, in his 1851 doctoral thesis, Riemann turned what had formerly been done on a case-by-case basis into a general theory. He proved that all bounded planar region were conformally equivalent, provided they had no holes in them. That is, any region with no holes (such regions are called "simply connected") could be mapped conformally to the the interior of a circle of radius 1, a "unit disk." Later mathematicians removed even the hypothesis of boundedness. The only simply connected planar region that cannot be conformally mapped to a unit disk is the plane itself.
Riemann's proof was controversial because of his assumption of what became known as the "Dirichlet principle." This principle could be summed up by the aphorism, "You can't fool Mother Nature." For example, no matter what distribution of electric charge you place on the boundary of a region, "Mother Nature" can find an electrostatic potential in the interior that matches the charge distribution. This potential turns out to be the real part of a complex-analytic function. To physicists this fact was self-evident; however, mathematicians considered it entirely too empirical to be used in a mathematical proof. Over the next 80 years, they found other ways to prove Riemann's celebrated theorem. Finally, however, David Hilbert vindicated Riemann by proving that the Dirichlet principle is true for the types of regions that Riemann was considering. It is no exaggeration to say that the entire field of elliptic partial differential equations has grown out of the quest to prove the Dirichlet principle.
The concept of conformal equivalence can be applied to curved surfaces, as well as planar regions. The "uniformization theorem" says that any closed surface with no boundary has one of only three types of conformal geometry. For example, the equation z2 + w22 = 1 (where z and w are complex, not real, variables) defines a surface that is conformally equivalent to a sphere. The equation z^2 = w^3 - w defines a torus, which has the same conformal geometry as a plane. Most polynomial equations involving exponents greater than three, when defined over the complex numbers, define toruses with two or more holes. These have the conformal geometry of a unit disk. The absence of other possibilities stems directly from the Riemann mapping theorem, because the geometry of the surface is first "lifted" to a planar covering space and then mapped by Riemann's theorem to the unit disk. In this way the theorem provides a great deal of information on the oldest problem of algebraic geometry, to link the geometry of a surface to the algebraic equation that defines it.
Other directions of research inspired by the Riemann mapping theorem include:
three-dimensional topology, where William Thurston has suggested a "Geometrization Conjecture" that classifies three-dimensional manifolds into eight distinct geometric types;
the theory of several complex variables, which classifies the conformal types of regions in two-dimensional complex space (four-dimensional Euclidean space) and higher;
and the theory of circle-packings, a computer-friendly method of computing the mappings that are guaranteed to exist by the Riemann mapping theorem.
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