S.: An optimal systolic inequality for CAT(0) metrics in genus two, preprint

Abstract. We prove an optimal systolic inequality for CAT(0) metrics on a genus 2 surface. We use a Voronoi cell technique, introduced by C. Bavard in the hyperbolic context. The equality is saturated by a flat singular metric in the conformal class defined by the smooth completion of the curve y2 = x5 − x. Thus, among all CAT(0) metrics, the one with the best systolic ratio is composed of six flat regular octagons centered at the Weierstrass points of the Bolza surface. Contents 1. Hyperelliptic surfaces of nonpositive curvature 1 2. Distinguishing 16 points on the Bolza surface 3 3. A flat singular metric in genus two 4 4. Voronoi cells and Euler characteristic