Deforming hyperbolic hexagons with applications to the arc and the Thurston metrics on Teichm{\"u}ller spaces

TL;DR

This paper constructs a family of hyperbolic hexagons with minimal Lipschitz maps and applies this to find new geodesics in the arc and Thurston metrics on Teichmüller spaces, advancing understanding of hyperbolic geometry.

Contribution

It introduces a novel method for deforming hyperbolic hexagons and demonstrates its application in identifying new geodesics in important Teichmüller metrics.

Findings

Constructed a one-parameter family of hyperbolic hexagons with minimal Lipschitz maps.

Discovered new geodesics for the arc metric on Teichmüller space of bordered surfaces.

Identified new geodesics for Thurston's metric on Teichmüller spaces of closed hyperbolic surfaces.

Abstract

For each right-angled hexagon in the hyperbolic plane, we construct a one-parameter family of right-angled hexagons with a Lipschitz map between any two elements in this family, realizing the smallest Lipschitz constant in the homotopy class of this map relative to the boundary. As a consequence of this construction, we exhibit new geodesics for the arc metric on the Teichm{\"u}ller space of an arbitrary surface of negative Euler characteristic with nonempty boundary. We also obtain new geodesics for Thurston's metric on Teichm{\"u}ller spaces of hyperbolic surfaces without boundary.