Shortening all the simple closed geodesics on surfaces with boundary

TL;DR

This paper proves that for surfaces with boundary, it is possible to find a hyperbolic metric that shortens all simple closed geodesics, with a uniform positive lower bound, advancing the understanding of Teichmüller space geometry.

Contribution

It provides a proof of Thurston's unpublished result on shortening geodesics on surfaces with boundary and establishes a uniform positive lower bound for the lengths.

Findings

Existence of a hyperbolic metric shortening all simple closed geodesics.

Shortening can be bounded below by a positive constant.

Improves upon recent results by Parlier.

Abstract

We give a proof of an unpublished result of Thurston showing that given any hyperbolic metric on a surface of finite type with nonempty boundary, there exists another hyperbolic metric on the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.