Thurston’s Metric on the Teichmüller Space of Flat Tori

We briefly review Thurston’s metric theory for complete hyperbolic surfaces with finite area, as well as some recent results about it. Then we review the generalization of Thurston’s metric theory for flat tori studied in Greenfield and Ji (Asian J Math 25(4):477–504, 2021) and Saǧlam (Int Electron J Geom 14(1):59–65, 2021) [13]. In particular, we review the connection between Lipschitz-extremal homeomorphisms and affine maps between flat tori and show that Thurston’s metric on the Teichmüller space of flat tori coincides with the Teichmüller metric.