Riemann mapping theorem

The Riemann mapping theorem is a fundamental result in complex analysis. It states that any non-empty, simply connected, open subset of the complex plane that is not the entire plane can be mapped conformally (angle-preserving) onto the open unit disk. This means there exists a bijective holomorphic function between the subset and the unit disk, with a holomorphic inverse. Such a mapping is called a conformal equivalence.

This theorem is important because it shows that all simply connected domains (except the whole plane) are essentially the same from a conformal geometry perspective. They can all be transformed into one another through conformal mappings. This has applications in various areas of mathematics and physics where preserving angles is crucial, such as fluid dynamics and electrostatics.

The theorem also leads to several deeper insights. For example, it implies that any simply connected domain in the plane is homeomorphic to the unit disk. The proof typically involves advanced concepts like normal families and extremal functions, connecting the theorem to broader areas of complex analysis and topology.

Sketch proof via Dirichlet problem

To understand how we can create a special kind of math rule that turns one area into a circle, we start with a special area called U and a point inside it called z₀. We want to build a math rule f that changes U into a unit disk (a circle with radius 1) and moves z₀ to the center of the circle, which is 0.

We can try a simple idea: use a rule that looks like (z – z₀) times e^g(z), where g(z) is another special math rule we need to find. This rule makes sure that z₀ is the only point that gets sent to the center. To make this work, we need to check that the size of our rule f(z) is exactly 1 when z is on the edge of U. This means we need to find a special math rule called u(z) that follows a pattern on the edge and spreads out evenly inside U. This is helped by a math idea called the Dirichlet principle, which tells us such a rule exists. Once we have this rule, we can find the rest of the pieces needed to finish our math rule f.

Uniformization theorem

The Riemann mapping theorem can be expanded to include Riemann surfaces. If you have a special area U on a Riemann surface that is simply connected and not empty, then U can be perfectly matched to one of three simple shapes: the Riemann sphere, the complex plane, or the unit disk. This idea is called the uniformization theorem.

Smooth Riemann mapping theorem

When a special kind of shape in complex numbers has smooth edges, the special mapping and its changes can be extended to the entire shape. This can be shown using ideas from studying solutions to certain math problems, either through Sobolev spaces for planar domains or classical potential theory. Other ways to prove this include using kernel functions or the Beltrami equation.

Algorithms

Computational conformal mapping is important in many areas, such as applied analysis, mathematical physics, and engineering, including image processing.

In the early 1980s, mathematicians discovered a simple way to compute conformal maps. This method can find a map between the unit disk and a region shaped like a closed loop, using a set of points on that loop. The method works well for many different shapes and gives good estimates of the map and its reverse.

Researchers have also studied how to use these maps to solve other difficult problems in computing, showing connections between conformal mapping and complex computational tasks.