Seminar Outline Dynamics and geometry

2. Teichmüller’s theorem: If X is compact, or more generally if X is obtained from a compact Riemann surface by removing a finite number of points, then there is an essentially map f minimizing the infimum above. (This map is unique up to conformal automorphisms of X and Y which are isotopic to the identity). The unique map is characterized by the fact that its Beltrami coefficient μ = ∂f/∂f is of the form μ = tφ/|φ|, where |t| < 1 and φ is a holomorphic quadratic differential on X with ∫