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

Abstract

Abstract. Let E be an infinite closed set in the Riemann sphere, and let T (E) denote its Teichmüller space. In this paper, we study some metric properties of T (E). We prove Earle's form of Teichmüller contraction for T (E), holomorphic isometries from the open unit disk into T (E), extend Earle's form of Schwarz's lemma for classical Teichmüller spaces to T (E), and finally study complex geodesics and unique extremality for T (E).