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

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

2005-01-02

arxiv.org/abs/math/0501017v2

Mathematics

Differential Geometry

14 pages, 1 figure, to appear in Pacific J. Math

Scientific paper

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.

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