An optimal systolic inequality for CAT(0) metrics in genus two

Mikhail Katz, Stéphane Sabourau
2006 Pacific Journal of Mathematics  
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 y 2 = x 5 − 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. MSC2000: 53C20, 53C23.
doi:10.2140/pjm.2006.227.95 fatcat:hrjnanqg6rfkpfwfijnabzk3ny