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

An old theorem of Harnack states that a symmetry of a compact Riemann surface X of genus g, (g ≥ 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 v(g) being the maximum number of ovals that a surface of genus g admits, v(g) ≤ 42(g − 1). We show in this paper that actually for g ≠ 2,3,5,7,9, v(g) ≤ 12(g − 1), that this bound is sharp for infinitely many g and we calculate v(g) for the mentioned above exceptional values of g as well.