Tracking a moving point in the plane
Frederick P. Gardiner, Nikola Lakic · Transactions of the American Mathematical Society · 2012
The Teichmüller theory of any hyperbolic Riemann surface R R induces two closely related metrics on R R in the following way. From a theorem of Bers, the fiber \[ K = Ψ − 1 ( [ i d e n t i t y ] ) \mathbb {K}= \Psi ^{-1}([identity]) \] of the forgetful map Ψ \Psi from the Teichmüller space T e i c h ( R − p ) Teich(R-p) onto the Teichmüller space T e i c h ( R ) Teich(R) is conformal to a disc and the evaluation map K ∋ [ f ] ↦ f ( p ) ∈ R \mathbb {K} i [f] \mapsto f(p) \in R is a universal covering of R . R. There are two infinitesimal metrics on K \mathbb {K} coming from Kobayashi’s construction: T e i c h K