#### Vol. 227, No. 1, 2006

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

### Mikhail G. Katz and Stéphane Sabourau

Vol. 227 (2006), No. 1, 95–107
##### 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 ${y}^{2}={x}^{5}-x$. 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.

##### Keywords
Bolza surface, CAT(0) space, hyperelliptic surface, Voronoi cell, Weierstrass point, systole
##### Mathematical Subject Classification 2000
Primary: 53C20, 53C23