About the math
The region can't be mapped onto the sphere minus a single point. By stereographic projection the sphere minus a point is the complex plane ℂ, and a conformal bijection ℂ → Ω onto a bounded region would be a bounded entire function. Liouville's theorem says such a function is constant.
What the Riemann mapping theorem gives you is a map f : Ω → 𝔻 onto the unit disk. It is unique once you fix f(z0) = 0 and f′(z0) > 0. The disk is conformally equivalent to the sphere minus a closed polar cap: |ζ| < R under inverse stereographic projection. So this page draws
Ω →f 𝔻 →·R {|ζ| < R} →σ⁻¹ S² ∖ cap
then rotates the sphere. Every setting is an honest conformal bijection. As the cap shrinks (R → ∞) the maps get closer to “sphere minus the north pole”, but they degenerate: everything outside a shrinking neighbourhood of the removed point is squeezed into a shrinking neighbourhood of z0, and almost the whole region shows only the area right around the cap. That squeeze is Liouville's theorem made visible.
f is computed in the browser with Marshall & Rohde's geodesic zipper algorithm, in double precision, spread across Web Workers. The boundary error in the readout is max | |f| − 1 | at the midpoints between boundary nodes. Drawing uses WebGL 2; if that is missing or the browser drops it (Safari does this when its graphics process runs out of resources), a software renderer takes over.
Imagery: NASA Earth Observatory, Blue Marble Next Generation (July 2004, Reto Stöckli). Map data: Natural Earth (public domain), which draws de facto control. The political map changes that in one respect: where UN resolutions declare a territorial change invalid, or the UN lists a territory as non-self-governing, the internationally recognised status is shown: Crimea → Ukraine (UNGA 68/262; drawn without its 2014 line), Golan Heights → Syria (UNSC 497), Northern Cyprus → Cyprus (UNSC 541), and Western Sahara as one territory (a UN non-self-governing territory; the Moroccan berm and the 27°40′N line are dashed). Other disputed, undemarcated or provisional boundaries and lines of control are dashed too (Kashmir, Aksai Chin, Arunachal, the Cyprus Green Line, the Golan lines, Korea, …), and Siachen Glacier, claimed by both India and Pakistan, is neutral grey.