Compact non-orientable surfaces of genus $4$ with extremal metric discs

A compact hyperbolic surface of genus $g$ is said to be extremal if it admits an extremal disc, a disc of the largest radius determined by $g$. We know how many extremal discs are embedded in a non-orientable extremal surface of genus $g=3$ or $g>6$. We show in the present paper that there exist $144$ non-orientable extremal surfaces of genus $4$, and find the locations of all extremal discs in those surfaces. As a result, each surface contains at most two extremal discs. Our methods used here are similar to those in the case of $g=3$.