Nonpositively curved surfaces are Loewner
Differential Geometry
2024-07-04 v2
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.
Cite
@article{arxiv.2404.00757,
title = {Nonpositively curved surfaces are Loewner},
author = {Mikhail G. Katz and Stephane Sabourau},
journal= {arXiv preprint arXiv:2404.00757},
year = {2024}
}
Comments
10 pages. To appear in Journal of Geometric Analysis