Asymptotic Teichmüller space of a closed set of the Riemann sphere

Abstract

The asymptotic Teichmüller space A T ( E ) AT(E) of a closed subset E E of the Riemann sphere C ^ \hat {\mathbb {C}} with at least 4 4 points and the natural asymptotic Teichmüller metric are introduced. It is proved that A T ( E ) AT(E) is isometrically isomorphic to the product space of the asymptotic Teichmüller spaces of the connected components of C ^ ∖ E \hat {\mathbb {C}}\setminus E and the Banach space of the Beltrami coefficients defined on E E . Furthermore, it is proved that there is a complex Banach manifold structure on A T ( E ) AT(E) .