Real structures of Teichmuller spaces, Dehn twists, and moduli spaces of real curves

An orientation reversing involution sigma of a topological compact genus g, g > 2, surface Sigma induces an antiholomorphic involution sigma* : T-g -- > T-g of the Teichmuller space of genus g Riemann surfaces. Two such involu tions sigma* and tau* are conjugate in the mapping class group if and only if the corresponding orientation reversing involutions sigma and tau of Sig ma are conjugate in the automorphism group of Sigma. This is equivalent to saying that the quotient surfaces Sigma/[sigma] and Sigma/[tau] are homeomo rphic. Hence the Teichmuller space T-g has m(g) = [3g+4/2] distinct antihol omorphic involutions, which are also called real structures of T-g ([7]). T his result is a simple fact that follows from Royden's theorem ([4]) statin g that the the mapping class group is the full group of holomorphic automor phisms of the Teichmuller space(g > 2). Let sigma* : T-g --> T-g and tau* : T-g --> T-g be two real structures that are not conjugate in the mapping c lass group. In this paper we construct a real analytic diffeomorphism d : T -g --> T-g such that sigma* = d(-1) circle tau* circle d. (1) This mapping d is a produce of full and half Dehn-twists around certain sim ple closed curves on the surface Sigma. This has applications to the moduli spaces of real algebraic curves. A compact Riemann surface (Sigma, X) admi tting an antiholomorphic involution sigma : (Sigma, X) --> (Sigma, X) is a real algebraic curve of the topological type Sigma/[sigma]. All fixed-point s of the real structure sigma* of the Teichmuller space T-g, are real curve s of the above topological type and every real curve of that topological ty pe is represented by an element of the fixed-point set of sigma*. The fixed -point set T-sigma*(g) is the Teichmuller space of real algebraic curves of the corresponding topological type. Given different real structures sigma* and tau*, let d the the real analytic mapping satisfying (1). It follows t hat d maps T-sigma*(g) onto T-tau*(g) and is an explicit real analytic diff eomorphism between these Teichmuller spaces.