2 8 N ov 2 00 3 On the number of extremal surfaces

Let X be a compact Riemann surface of genus ≥ 2 of constant negative curvature −1. An extremal disk is an embedded (resp. covering) disk of maximal (resp. minimal) radius. A surface containing an extremal disk is an extremal surface. This paper gives formulas enumerating extremal surfaces of genus ≥ 4 up to isometry. We show also that the isometry group of an extremal surface is always cyclic of order 1, 2, 3 or 6. Introduction Let X be a compact Riemann surface of genus ≥ 2 of constant negative curvature −1. We consider the maximal radius of an embedded metric disk in X and the minimal radius of a disk covering X. An extremal disk is an embedded (resp. covering) disk of maximal (resp. minimal) radius. A surface containing an extremal disk is an extremal surface. C.Bavard [1] proved, that if a surface contains an embedded disk of maximal radius if and only if it contains a covering disk of minimal radius and that extremal surfaces are modular surfaces. The radius Rg of an extremal embeddes disk, as well as the radius Cg of an extremal covering disk were computed in [1]: Rg = cosh (1/2sinβg), βg = π/(12g − 6). Cg = cosh (1/ √ 3tanβg), βg = π/(12g − 6).