While preparing for my presentation, I encountered a challenge in providing a rigorous elementary proof for the assertion:
The thrice-punctured sphere admits a unique complete finite-area hyperbolic metric, up to isometry.
Here is my argument:
Euler Characteristic: The Euler characteristic of a thrice-punctured sphere is $-1$. According to the uniformization theorem, this implies that a complete hyperbolic metric can be defined on the thrice-punctured sphere.
Ideal Triangles: In hyperbolic geometry, any two ideal triangles are isometric. Ideal triangles are special triangles where all three vertices lie on the boundary of the hyperbolic plane, and their sides are geodesics.
Gluing: By naturally gluing two ideal triangles together, we can construct a complete finite-area hyperbolic thrice-punctured sphere.
Uniqueness: Conversely, starting with a complete finite-area hyperbolic thrice-punctured sphere, we can cut it along three long seams, which are simple bi-infinite geodesics connecting two distinct cusps. This operation results in two ideal triangles.
In reviewing this argument, please verify if there are any gaps or ambiguities. Additionally, if you can recommend any references or sources that provide a complete proof, I would greatly appreciate it.