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.

Keywords

Bolza surface, CAT(0) space, hyperelliptic surface, Voronoi cell, Weierstrass point, systole

Mathematical Subject Classification

Primary: 53C20, 53C23

Authors