Extremal disks and extremal surfaces of genus three

抄録

A compact Riemann surface of genus g≥2 is said to be extremal if it admits an extremal disk, a disk of the maximal radius determined by g. If g=2 or g≥4, it is known that how many extremal disks an extremal surface of genus g can admit. In the present paper we deal with the case of g=3. Considering the side-pairing patterns of the fundamental polygons, we show that extremal surfaces of genus 3 admit at most two extremal disks and that 16 surfaces admit exactly two. Also we describe the group of automorphisms and hyperelliptic surfaces.