Maximal stretch and Lipschitz maps on Riemannian manifolds of negative curvature

TL;DR

This paper extends Thurston's concept of maximal stretch from hyperbolic surfaces to negatively curved Riemannian manifolds, exploring the structure of the Mather set and its relation to Lipschitz maps.

Contribution

It introduces a generalized notion of maximal stretch for negatively curved manifolds and analyzes the Mather set, highlighting similarities and differences with classical geodesic laminations.

Findings

Mather set may not be lifts of geodesic laminations in this setting

Connections established between Mather set and best Lipschitz maps

Structural properties of the Mather set analyzed in negatively curved manifolds

Abstract

In his seminal work on Teichm\"uller spaces (\cite{Th98}), Thurston introduced the maximal stretch for a pair of hyperbolic metrics on a closed surface of genus $\geq 2$ and showed that the logarithm of this quantity induces an asymmetric metric in the Teichm\"uller space. He also showed that the subset of the surface on which the maximal stretch is attained is a geodesic lamination. In this paper, we define the maximal stretch analogously for closed manifolds equipped with Riemannian metrics of variable negative curvature and investigate the structure of the related Mather set on the unit tangent bundle. In contrast to the Teichm\"uller space, the Mather set may not be lifts of geodesic laminations in this broader setting. However, in our paper, we will discuss similar features shared by the Mather set with geodesic laminations. We also connect the study of the Mather set with the theory of best Lipschitz maps.