# Uniqueness of distance realizing geodesic in hyperbolic surface. [duplicate]

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Hyperbolic surfaces

Given a hyperbolic surface S with geodesic boundary. Let a and b be two distinct simple closed geodesic boundaries. Does there exist a unique distance realizing geodesic in S? (1) for S is a pair of pants. (2) S is any hyperbolic surface with boundary.

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## marked as duplicate by S. Carnahan♦May 2 '12 at 7:15

They are always disjoint. This is proved by the usual "exchange and round-off" technique. See, if $p$ and $q$ intersect, then at the intersection point you can do a little cut and paste to get two new arcs $p'$ and $q'$, still connecting the same boundary components, and slightly shorter, which is a contradiction. –  Sam Nead Jan 3 '14 at 11:00