Nonpositively Curved Surfaces are Loewner

Abstract

We show that every closed nonpositively curved surface satisfies Loewner’s systolic inequality. The proof relies on a combination of the Gauss–Bonnet formula with an averaging argument using the invariance of the Liouville measure under the geodesic flow. This enables us to find a disk with large total curvature around its center yielding a large area.