On Teichmueller's metric and Thurston's asymmetric metric on Teichmueller space

Athanase Papadopoulos (1) , Guillaume Théret (1)

1 IRMA - Institut de Recherche Mathématique Avancée

We review some elements of Teichmueller's metric and of Thurston's asymmetric metric on Teichmueller space. One of our objectives is to draw a parallel between these two metrics and to stress on some differences between them. The results that we present on Teichmueller's metric are classical, whereas some of the results on Thurston's asymmetric metric are new. We also discuss some open problems.

Keywords: conformal structure; Riemann surface; hyperbolic structure; Teichmueller space; Teichmueller metric; Teichmueller geodesic; Thurston's asymmetric metric; extremal length; modulus; quasiconformal homeomorphism; measured foliation

Domains: Complex Variables [math.CV]

Athanase Papadopoulos, Guillaume Théret. On Teichmueller's metric and Thurston's asymmetric metric on Teichmueller space. A. Papadopoulos. Handbook of Teichmüller Theory, Volume 1, 11, European Mathematical Society Publishing House, 2007, IRMA Lectures in Mathematics and Theoretical Physics, ⟨10.4171/029-1/3⟩. ⟨hal-00129729⟩