Compact non-orientable surfaces of genus 6 with extremal metric discs

Abstract

A compact hyperbolic surface of genus g g is said to be extremal if it admits an extremal disc, a disc of the largest radius determined only by g g . We discuss how many extremal discs are embedded in non-orientable extremal surfaces of genus 6. This is the final genus in our interest because it is already known for g = 3 , 4 , 5 g=3, 4, 5 , or g > 6 g>6 . We show that non-orientable extremal surfaces of genus 6 admit at most two extremal discs. The locus of extremal discs is also obtained for each surface. Consequently non-orientable extremal surfaces of arbitrary genus g ≧ 3 g\geqq 3 admit at most two extremal discs. Furthermore we determine the groups of automorphisms of non-orientable extremal surfaces of genus 6 with two extremal discs.