Shortening all the simple closed geodesics on surfaces with boundary

Details Shortening all the simple closed geodesics on surfaces with boundary Shortening all the simple closed geodesics on surfaces with boundary

2009-05-28

arxiv.org/abs/0905.4661v2

Mathematics

Geometric Topology

Revised version, to appear in the Proceedings of the AMS

Scientific paper

We give a proof of an unpublished result of Thurston showing that given any hyperbolic metric on a surface of finite type with nonempty boundary, there exists another hyperbolic metric on the same surface for which the lengths of all simple closed geodesics are shorter. (This is not possible for surfaces of finite type with empty boundary.) Furthermore, we show that we can do the shortening in such a way that it is bounded below by a positive constant. This improves a recent result obtained by Parlier in [2]. We include this result in a discussion of the weak metric theory of the Teichm\"uller space of surfaces with nonempty boundary.

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