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

Athanase Papadopoulos; Sumio Yamada

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arXiv · 1512.02776

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

Athanase Papadopoulos·Sumio Yamada

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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.

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BibTeX RIS

Athanase Papadopoulos, Sumio Yamada. 2015-12-09. Deforming hyperbolic hexagons with applications to the arc and the Thurston metrics on Teichm{\"u}ller spaces. https://arxiv.org/abs/1512.02776

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