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An optimal systolic inequality for \(\mathrm{CAT}(0)\) metrics in genus two. (English) Zbl 1156.53019
The systole of a Riemannian manifold is the shortest length of closed noncontractible loops, and when the dimension is equal to 2, the systolic ratio is the square of the systole divided by the area of the surface. It is invariant under scaling. An important problem is to determine the optimal systolic ratio for natural classes of surfaces. For the Klein bottle (the real projective plane) and the torus, it was determined by Pu and Loewner respectively. Such an optimal bound is not known for higher genus surfaces.
This paper gives an optimal systolic ratio inequality for CAT(0) metrics on a genus 2 surface with mild singularities (i.e., conic singularities whose angles at the singular points are rational multiples of \(2\pi\)), and shows that the optimal bound is realized by a singular metric, with 16 conical singular points, in the conformal class of the Bolza surface. It uses a Voronoi cell technique introduced by C. Bavard in the hyperbolic context to prove the results.

53C20 Global Riemannian geometry, including pinching
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
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