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

2006/09/01 by Mikhail G. Katz, Stéphane Sabourau · 1 citation
Mathematics · #Combinatorics #Conformal map #Context (archaeology) #Genus #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry #Geometry and complex manifolds #Mathematical analysis #Mathematics #Metric (unit) #Surface (topology) #Voronoi diagram

paper · pdf · doi:10.2140/pjm.2006.227.95

openalex publication_date 2006/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/22

Abstract

Abstract. We prove an optimal systolic inequality for J-invariant, CAT(0) metrics on a genus 2 surface, where J is the hyperelliptic involution. We use a Voronoi cell technique, introduced by C. Bavard in the hyperbolic context. The boundary case of equality is satisfied 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 J-invariant 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. Contents 1. Hyperelliptic surfaces of nonpositive curvature 1 2. Systolic ratios 3 3. Distinguishing 16 points on the Bolza surface B 3

Citations

• Hyperelliptic surfaces are Loewner
• Metric Structures for Riemannian and Non-Riemannian Spaces
• Systolic Geometry and Topology
• Algebraic Curves and Riemann Surfaces
• Filling Riemannian manifolds
• Some inequalities in certain nonorientable Riemannian manifolds
• E7, Wirtinger inequalities, Cayley 4-form, and homotopy
• Reimann surfaces with shortest geodesic of maximal length
• Entropy of systolically extremal surfaces and asymptotic bounds
• Metric Spaces of Non-Positive Curvature
• Riemannian Geometry

Cited by

• Bolza surface [wikipedia]