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.