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

Mikhail G. Katz and Stéphane Sabourau

Vol. 227 (2006), No. 1, 95–107

DOI: 10.2140/pjm.2006.227.95

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

Milestones

Received: 13 October 2004

Accepted: 10 April 2006

Published: 1 September 2006