Thrice-punctured sphere

Question

This claim is made in the book Quantum Triangulations (eds.: Carfora, Marzuoli), p.45:

the thrice-punctured sphere is the largest subdomain of $\mathbb{S}^2$ supporting a hyperbolic metric.

I would appreciate it if someone could either explain this statement, or point me to the appropriate source that would help me understand it. Thanks!

Answer 1

A pair of pants, as a (subset of a) thrice-punctured sphere, admits a hyperbolic structure, unlike the unpunctured or once or twice punctured spheres (sphere, plane, annulus), which admit positive curvature, zero curvature, and zero curvature, respectively – compare Little Picard theorem.

http://en.wikipedia.org/wiki/Pair_of_pants_(mathematics)

See page 3 of Buser and Parlier and Kleinian

Starting with the thrice-punctured sphere, given its unique complete hyperbolic metric, one can construct punctured tori by cutting off horoball neighborhoods of two cusps and gluing their circular boundaries by some möbius transformation. The result has a natural complex projective (möbius) structure, whose underlying conformal structure can be uniformized to a hyperbolic structure.