Nonpositively curved surfaces are Loewner
arXiv:2404.00757 · doi:10.1007/s12220-024-01732-4
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.
10 pages. To appear in Journal of Geometric Analysis