Theorem. Riemann’s mapping theorem [rmsf-1407][edit]

$\gdef\CC{\mathbb{C}}$

Let $U \in \CC$ be an open subset which is simply-connected and not equal to $\CC$. Then there exists a biholomorphism between $U$ and the unit disk $D = D(0,1)$.

We will explain the term “simply-connected” later.

There is something of a physical “proof” of this theorem, which also give some intuition for the notion of a biholomorphism. The idea is this: we can decorate the unit disk by imagining the radial lines joining the center $0$ to the boundary, and also imagining all the circles of radius $< 1$ centered at $0$. We want to find the corresponding decorations of $U$; then the biholomorphism will be constructed to match these up.

For this, we start by fixing a point $p \in U$ which will correspond to $0 \in D$. Then we place an electric charge at $p$, and consider the magnetic field it generates. But we do this subject to the boundary condition that the potential should be zero on the boundary of $U$. Then we can decorate $U$ by the field lines on the one hand, and by the equipotential lines on the other. These will correspond to the radial lines and the circles in $D$.

The reason this all works, heuristically, is that field lines are always perpendicular to equipotential lines. That is why the resulting map $U \lrarr D$ will be biholomorphism: right angles are preserved.

That was of course a non-rigorous argument, but one can make it rigorous. At the very least, there is a moral to the story: since nature decides what to do by solving real differential equations, we should try to construct holomorphic functions using solution to (certain) real differential equations as well.