Spaces of Riemann Surfaces as Bounded Domains by Lipman Bers

1. A surface of type (g, n) is a surface 5 obtained by removing n*zO distinct points from a closed Riemann surface of genus g\ we assume always that 3g — 3+n>0. Such a surface 5 is marked by choosing a homotopy class [ƒ] of orientation preserving homeomorphisms ƒ of 5 onto a fixed reference surface So. Two marked surfaces (Si, [ft]) a n d (S2, [ƒ2]) are equivalent if there is a conformai mapping g with g(S\) = S2, [/2g] = \fi]. The Teichmilller space Tgtn is the set of equivalence classes of marked surfaces of type (g, n). I t is known that TQtn carries two natural structures: that of a metric space, homeomorphic to a (6g — 6+2w)-cell [ l ; 4; 10; 11 ], and that of a complex manifold [2; 5; 7; 8; 12]. We sketch in §§2-7 a proof of the