Sharp systolic bounds on negatively curved surfaces (Q6540602)

The main result of this paper is that every local supremum of the systole over the space of Riemannian metrics of curvature \(\leq -1\) on a given non-simply connected closed surface is attained at a hyperbolic metric. The authors also discuss the metric setting of surfaces with Alexandrov curvature. They obtain in this case a result which holds for local suprema of the systole. As a corollary they obtain that for any non-simply connected closed surface, the systole function on the space of Alexandrov surfaces of curvature at \(\leq -1\) has only finitely many local suprema.