Why is conformal mapping important for Riemann Sphere

Question

The mapping of the complex plane to the Riemann sphere is conformal. As a result, many sources contend that it's this property of conformal that makes the concept of Riemann Sphere non-trivial. Hencewhy, I wonder what's the reason behind this. There are a few aspects that I would love to see from the answers. 1. How the conformal property of the Riemann Sphere is related to holomorphic functions (I know that Holomorphic function is basically conformal mapping, but I expect more details). 2. Circles get mapped to circles, the infinitesimal shape is preserved, and angles are preserved locally. Why are these properties important, and what applications are implied?

Answer 1

By definition, a smooth map is conformal if its real differential preserves angles.

It is a theorem that a map defined on say $\mathbb C \simeq \mathbb R^2$ is conformal if and only if it is holomorphic and its derivative doesn't vanish. Note that a conformal map defined on some open set of the Riemann sphere does not in general map circles to circles, it's just its derivative that does (this is what is meant by "maps infinitesimal circles to infinitesimal circles").

Maps that are defined and conformal on the whole Riemann sphere are bijective and their inverse is also conformal. They can be shown to be exactly the maps of the form $f(z)=\frac{az+b}{cz+d}$, which are called Möbius transformations. They maps circles or lines to circles or lines. They are important because they are the automorphisms of the Riemann sphere, which basically means that any complex analysis problem in the Riemann sphere can be considered "up to a Möbius transformation". For example, since there is a Möbius transformation mapping the upper half-plane $\{z \in \mathbb C: \mathrm{Im} z >0\}$ to the unit disk $\{ z \in \mathbb C : |z|<1\}$, these two sets can be used interchangeably in many situations.