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Some Metric Properties of the Teichmüller Space of a Closed Set in the Riemann Sphere by Nishan Chatterjee Advisors: Yunping Jiang and Sudeb Mitra Associated to each closed subset E of the Riemann sphere Ĉ, there is a contractible complex Banach manifold with a basepoint; this was first studied by G. Lieb in his Cornell University doctoral dissertation [15]. We call this the Teichmüller space of the closed set E, denoted by T (E). Throughout this thesis, the blanket assumption will be that E is a closed set in Ĉ and that 0, 1,∞ belong to E. The Teichmüller space T (E) is intimately related with holomorphic motions of the closed set E. In this thesis we study several metric and analytic properties of T (E). For the Teichmüller space of a Riemann surface, the principle of Teichmüller contraction was introduced by Gardiner in the paper [9]. In [18] Mitra proved a δform of Teichmüller contraction for the generalized Teichmüller space T (E). Our first theorem in this thesis, is to extend Earle’s form of Teichmüller contraction

Some metric properties of the Teichmüller space of a closed set in the Riemann sphere