Compact non-orientable hyperbolic surfaces with an extremal metric disc

Abstract

The size of a metric disc embedded in a compact non-orientable hyperbolic surface is bounded by some constant depending only on the genus g ≥ 3 g \ge 3 . We show that a surface of genus greater than six contains at most one metric disc of the largest radius. For the case g = 3 g=3 , we carry out an exhaustive study of all the extremal surfaces, finding the location of every extremal disc inside them.