Genus of two extremal surfaces: Extremal discs, isometries and Weierstrass points (Q1860707)

The paper's abstract: ``It is known that the largest disc that a compact hyperbolic surface of genus \(g\) may contain has radius \(R=\cosh^{-1}(1/2 \sin (\pi/(12g-6)))\). It is also known that the number of such (extremal) surfaces, although finite, grows exponentially with \(g\). Elsewhere the authors have shown that for genus \(g>3\) extremal surfaces contain only one extremal disc. Here we describe in full detail the situation in genus 2. Following results that go back to Fricke and Klein we first show that there are exactly nine different extremal surfaces. Then we proceed to locate the various extremal discs that each of these surfaces possesses as well as their set of Weierstrass points and group of isometries. The number of sides of the symmetric fundamental polygon if \(g=2\) is 18, and all possible side pairings were worked out by \textit{R. Fricke}, \textit{E. Klein} [Vorlesungen über die Theorie der automorphen Funktionen (1897)]. Here the careful analysis of the geometry and trigonometry in each cases, the authors specify the locations of the centers of extremal discs and Weierstrass points. They are also charmingly illustrated.