Compact Klein surfaces of genus 5 with a unique extremal disc (Q2841086)

Let \(S\) be a non-orientable compact hyperbolic surface of genus \(g\). Then the radius \(r\) of a disc embedded in \(S\) satisfies the inequality NEWLINE\[NEWLINE \cosh r \leq \frac{1}{2 \sin \frac{\pi}{6(g-1)}}. NEWLINE\]NEWLINE Call \(R_g\) the radius satisfying the equality. The surface \(S\) is called extremal if a disc of radius \(R_g\), called an extremal disc, is isometrically embedded in \(S\).NEWLINENEWLINEThe author proved in [Glasg. Math. J. 54, No. 2, 273--281 (2012; Zbl 1258.30019)] that there are \(3627\) non-orientable extremal surfaces of genus 5. Seventeen among them have two extremal discs, and all the others have just one.NEWLINENEWLINEIn the paper under review the author obtains the automorphism group of each of the \(3610\) non-orientable extremal surfaces of genus 5 having a unique extremal disc. This group is \(D_3\) for \(4\) surfaces, \(\mathbb{Z}_3\) for \(2\) surfaces, \(\mathbb{Z}_2\) for \(402\) surfaces, and \(\{1\}\) for the remaining \(3202\) surfaces.