  Let $X$ be a compact Riemann surface of genus $\geq 2$ 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 {\em
extremal surface}.
  This paper gives formulas enumerating extremal surfaces of genus $\geq 4$ up
to isometry. We show also that the isometry group of an extremal surface is
always cyclic of order 1, 2, 3 or 6.
