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

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.

Bolza surface, CAT(0) space, hyperelliptic surface, Voronoi cell, Weierstrass point, systole
Mathematical Subject Classification 2000
Primary: 53C20, 53C23
Received: 13 October 2004
Accepted: 10 April 2006
Published: 1 September 2006
Mikhail G. Katz
Department of Mathematics
Bar Ilan University
Ramat Gan 52900
Stéphane Sabourau
Laboratoire de Mathématiques et Physique Théorique
Université de Tours
Parc de Grandmont
37400 Tours