Geometric characterization for affine mappings and Teichmüller mappings
We characterize affine mappings on the unit disk and on rectangles by module conditions. The main result generalizes the classic Schwarz lemma. As an application, we give a sufficient condition for a K-quasiconformal mapping on a Riemann surface to be a Teichmüller mapping. 1. Preliminaries. Let Γ be a family of curves in the plane. Each γ ∈ Γ is a countable union of open arcs, closed arcs or closed curves, and every closed subarc is rectifiable. We shall define the extremal length λ(Γ ) of Γ . A function %, defined on the whole plane, is called admissible if the following conditions are satisfied: (i) % is a non-negative Borel function, (ii) A(%) = C % 2 dx dy 6= 0,∞. If such a % is measurable as a function of arc length on γ, set (1.1) Lγ(%) = γ % |dz|. Otherwise, set Lγ(%) =∞. Define L(%) = inf γ∈Γ Lγ(%), (1.2) λ(Γ ) = sup % L(%)2 A(%) , (1.3) where the supremum is taken over all admissible %. We write Γ1 < Γ2 if for every γ2 ∈ Γ2 there is a γ1 ∈ Γ1 which is a subarc of Γ2. By the above definitions, the extremal length is monotonic: 2000 Mathematics Subject Classification: 30C62, 30C55, 30F60.