On a Harnack-Natanzon theorem for the family of real forms of Riemann surfaces

An old theorem of Harnack states that a symmetry of a compact Riemann surface X of genus g, (g greater than or equal to 2) has at most g + 1 disjoint simple closed curves of fixed points, each of which is called the oval of X. Much more recently Natanzon proved that for nu(g) being the maximum number of ovals that a surface of genus g admits, nu(g) less than or equal to 42(g-1). We show in this paper that actually for g not equal 2, 3, 5, 7, 9, nu(g) less than or equal to 12(g-1), that this bound is sharp for infinitely many g and we calculate nu(g) for the mentioned above exceptional values of g as well. (C) 1997 Elsevier Science B.V.