Rectifiability in Teichmüller theory

Abstract

1. The universal Teichmuller space. A simply connected domain Ω of the plane is well parametrized by its Riemman mapping, i.e. a conformal mapping from the unit disk ∆ onto Ω. The idea of universal Teichmuller space (UTS for short) is to equip the set of all those conformal mappings with an (infinite dimensional) complex structure, in order to study deformations of such domains. In order to guess what is this complex structure, let us first define a holomorphic deformation of the unit disk as a family fλ, λ ∈ ∆ of conformal mappings ∆→ C (i.e. holomorphic and injective) such that: (i) f0(z) = z, (ii) ∀z ∈ ∆, λ 7→ fλ(z) is holomorphic in ∆. The complex structure on UTS should be of course such that, when restricted to a holomorphic deformation of the disk it coincides with the λ-structure. Now, by the Koebe distortion theorem, for every Riemann mapping f on the unit disk, the function Log f ′ belongs to the Banach space B of holomorphic functions b in ∆ satisfying ‖b‖B = sup z∈∆ (1− |z|)|b′(z)|

Locations

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