Metrics and compactifications of Teichmüller spaces of flat tori

2019/03/26 by Greenfield, Mark, Ji, Lizhen
#30F60 #53C35 #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1903.10655

Abstract

Using the identification of the symmetric space SL(n,ℝ)/SO(n) with the Teichmüller space of flat n-tori of unit volume, we explore several metrics and compactifications of these spaces, drawing inspiration both from Teichmüller theory and symmetric spaces. We define and study analogs of the Thurston, Teichmüller, and Weil-Petersson metrics. We show the Teichmüller metric is a symmetrization of the Thurston metric, which is a polyhedral Finsler metric, and the Weil-Petersson metric is the Riemannian metric of SL(n,ℝ)/SO(n) as a symmetric space. We also construct a Thurston-type compactification using measured foliations on n-tori, and show that the horofunction compactification with respect to the Thurston metric is isomorphic to it, as well as to a minimal Satake compactification.

Related

• On the Finsler stucture of the Teichmüuller metric and Thurston's asymmetric metric
• The geometry of the Thurston metric: a survey
• Tangent spaces of the Teichmüller space of the torus with Thurston's weak metric
• Thurston's metric on Teichmüller space and isomorphisms between Fuchsian groups
• Thurston's metric on Teichmüller space of semi-translation surfaces