An optimal systolic inequality for CAT(0) metrics in genus two
Differential Geometry
2007-05-23 v2 Geometric Topology
Metric Geometry
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.
Cite
@article{arxiv.math/0501017,
title = {An optimal systolic inequality for CAT(0) metrics in genus two},
author = {Mikhail G. Katz and Stephane Sabourau},
journal= {arXiv preprint arXiv:math/0501017},
year = {2007}
}
Comments
14 pages, 1 figure, to appear in Pacific J. Math