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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

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

If you want to refine your question, you should edit the first one. There is an "edit" link below the tags. – S. Carnahan May 2 '12 at 7:16
up vote 1 down vote accepted

For the pants, yes. In general, no. To prove this for the pants, classify all geodesic arcs and just observe the result. There are many ways to find a "no" example in the general case; the first one that came to my mind was taking a double cover.

EDIT - I see that this is a near-duplicate of a closed question. You could improve your question by giving some motivation. Reading the FAQ will be very useful in writing questions that get good answers. In particular please see

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well, I have got an example of hyperbolic surface with boundary where more than one (at least two) distance realizing geodesics between two distinct geodesic boundaries will exist. I have a further question: Suppose p and q are two distance realizing geodesics between the boundary geodesics. Is it true that p and q are always disjoint? – Bidyut Sanki May 2 '12 at 8:26
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

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