How to understand the Riemann surfaces described by Ahlfors complex analysis

Question

I just can not understand how the author build the riemann surfaces. The details are following picture:

I searched on Google, read the definition of the Riemann surface and can understand the usual definition for it. But I cannot understand the construction of the Riemann surfaces by this book especially the different sheet and cut, etc.

Answer 1

The way to understand Riemann surfaces is to start with the square root function and go no further until mastering it. And begin with Riemann's original notion: create a new coordinate surface such that a multi-valued function becomes a single-valued function of the points on this new surface. The new surface is the Riemann surface. Now suppose we wanted to do just that without any regards to the notion of a "Riemann surface".

Consider the function $w=z^2=r^2 e^{2i\theta}$. That is a mapping of the entire $z$-plane to two copies of the $w$ plane is it not? Consider $r^2 e^{2i\theta}$ for $-\pi \leq \theta <0$. This maps one-to-one, the lower $z$-plane to all of the $w$-plane. Likewise, the upper $z$-plane maps to another copy of the $w$-plane. We can label these $w$-plane copies $k=1$ and $k=2$ or more commonly, $k=0$ and $k=1$. We can identify every point in each copy of the two $w$-planes by $(w,\sqrt{w})$. For example, copy $k=0$ would have the point $(1,1)$ and copy $k=1$ would have the point $(1,-1)$. So that every point $(u,v)$ in this $2$-copy $w$-plane we call a Riemann surface is unique and further, there is a one-to-one mapping of the points $(u,v)$ in the Riemann surface to the $z$-plane: $$\displaystyle f(w,r^{1/2}e^{i/2(\theta+2k\pi)})= r^{1/2}e^{i/2(\theta+2k\pi)}$$

Now we would like to make this Riemann surface continuous. We can easily do this by making a radial cut anywhere on each surface but most commonly along the negative real axis (or positive in Ahlfor's construction) and then splice the lower half of sheet $k=0$ to the upper half of sheet $k=1$ along the respective cut edges, and likewise, splice the upper half of sheet $k=0$ to the lower half of $k=1$ along those cut edges. So that as we move along this new coordinate system (or surface) say $w=2 e^{i\theta}$ from $\theta=\pi-0.1$ to $\pi+0.1$ we smoothly move across sheet $k=0$ to $k=1$ as moving from the red sheet to the blue sheet in the figures below.

The function $w=z^3$ follows naturally right? It's just three sheets spliced together. Try to edit the Mathematica code below to create a plot.

fun1[z_, n_] := Abs[z]^(1/2) Exp[I/2 (Arg[z] + 2 n 
Pi)];
sheet1 = ParametricPlot3D[{Re[z], Im[z], Im@fun1[z, 
0]}z -> r Exp[I t], {r, 0.1, 2}, {t, -Pi, Pi}, 
PlotStyle -> Red];
sheet2 = ParametricPlot3D[{Re[z], Im[z], Im@fun1[z, 
1]} /.z -> r Exp[I t], {r, 0.1, 2}, {t, -Pi, Pi}, 
PlotStyle -> Blue];
sheet3 = Show[{sheet1, sheet2}]