Compact non-orientable surfaces of genus 6 with extremal metric discs (Q2814487)

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\[NEWLINE \cosh r \leq \frac{1}{2 \sin \frac{\pi}{6(g-1)}}. NEWLINE\]NEWLINENEWLINENEWLINECall \(R_g\) the radius satisfying 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 and E. Girondo proved that extremal surfaces of genus \(g > 6\) admit a unique extremal disc, whilst those of genus \(3\) admit at most two extremal discs. For genera \(4\) and \(5\), the author proved also that there are at most two extremal discs in a given extremal surface.NEWLINENEWLINEThe paper under review completes the landscape, by considering genus \(6\). The main result is that there are 149279 non-orientable extremal surfaces of genus \(6\), up to isomorphism. They admit at most two extremal discs, and there are 107 with exactly two extremal discs. These 107 surfaces with two extremal discs are carefully studied, as well as their automorphism group. This group is \(\{1\}\) for 8 surfaces, \(\mathbb{Z}_2\) for 38 surfaces, \(\mathbb{Z}_2 \times \mathbb{Z}_2\) for 57 surfaces, \(D_3\) for 2 surfaces, and finally \(D_6\) for 2 surfaces.